# Neat Randomized Algorithms: RandDiag for Rapidly Diagonalizing Normal Matrices

Consider two complex-valued square matrices and . The first matrix is Hermitian, being equal to its conjugate transpose . The other matrix is non-Hermitian, . Let’s see how long it takes to compute their eigenvalue decompositions in MATLAB:

>> A = randn(1e3) + 1i*randn(1e3); A = (A+A')/2;
>> tic; [V_A,D_A] = eig(A); toc % Hermitian matrix
Elapsed time is 0.415145 seconds.
>> B = randn(1e3) + 1i*randn(1e3);
>> tic; [V_B,D_B] = eig(B); toc % non-Hermitian matrix
Elapsed time is 1.668246 seconds.

We see that it takes longer to compute the eigenvalues of the non-Hermitian matrix as compared to the Hermitian matrix . Moreover, the matrix of eigenvectors for a Hermitian matrix is a unitary matrix, .

There are another class of matrices with nice eigenvalue decompositions, normal matrices. A square, complex-valued matrix is normal if . The matrix of eigenvectors for a normal matrix is also unitary, . An important class of normal matrices are unitary matrices themselves. A unitary matrix is always normal since it satisfies .

Let’s see how long it takes MATLAB to compute the eigenvalue decomposition of a unitary (and thus normal) matrix:

>> U = V_A;                     % unitary, and thus normal, matrix
>> tic; [V_U,D_U] = eig(U); toc % normal matrix
Elapsed time is 2.201017 seconds.

Even longer than it took to compute an eigenvalue decomposition of the non-normal matrix ! Can we make the normal eigenvalue decomposition closer to the speed of the Hermitian eigenvalue decomposition?

Here is the start of an idea. Every square matrix has a Cartesian decomposition:

We have written as a combination of its Hermitian part and times its skew-Hermitian part . Both and are Hermitian matrices. The Cartesian decomposition of a square matrix is analogous to the decomposition of a complex number into its real and imaginary parts.

For a normal matrix , the Hermitian and skew-Hermitian parts commute, . We know from matrix theory that commuting Hermitian matrices are simultaneously diagonalizable, i.e., there exists such that and for diagonal matrices and . Thus, given access to such , has eigenvalue decomposition

Here’s a first attempt to turn this insight into an algorithm. First, compute the Hermitian part of , diagonalize , and then see if diagonalizes . Let’s test this out on a example:

>> C = orth(randn(2) + 1i*randn(2)); % unitary matrix
>> H = (C+C')/2;                     % Hermitian part
>> [Q,~] = eig(H);
>> Q'*C*Q                            % check to see if diagonal
ans =
-0.9933 + 0.1152i  -0.0000 + 0.0000i
0.0000 + 0.0000i  -0.3175 - 0.9483i

Yay! We’ve succeeded at diagonalizing the matrix using only a Hermitian eigenvalue decomposition. But we should be careful about declaring victory too early. Here’s a bad example:

>> C = [1 1i;1i 1]; % normal matrix
>> H = (C+C')/2;
>> [Q,~] = eig(H);
>> Q'*C*Q           % oh no! not diagonal
ans =
1.0000 + 0.0000i   0.0000 + 1.0000i
0.0000 + 1.0000i   1.0000 + 0.0000i

What’s going on here? The issue is that the Hermitian part for this matrix has a repeated eigenvalue. Thus, has multiple different valid matrices of eigenvectors. (In this specific case, every unitary matrix diagonalizes .) By looking at alone, we don’t know which matrix to pick which also diagonalizes .

He and Kressner developed a beautifully simple randomized algorithm called RandDiag to circumvent this failure mode. The idea is straightforward:

1. Form a random linear combination of the Hermitian and skew-Hermitian parts of , with standard normal random coefficients and .
2. Compute that diagonalizes .

That’s it!

To get a sense of why He and Kressner’s algorithm works, suppose that has some repeated eigenvalues and has all distinct eigenvalues. Given this setup, it seems likely that a random linear combination of and will also have all distinct eigenvalues. (It would take a very special circumstances for a random linear combination to yield two eigenvalues that are exactly the same!) Indeed, this intuition is a fact: With 100% probability, diagonalizing a Gaussian random linear combination of simultaneously diagonalizable matrices and also diagonalizes and individually.

MATLAB code for RandDiag is as follows:

function Q = rand_diag(C)
H = (C+C')/2; S = (C-C')/2i;
M = randn*H + randn*S;
[Q,~] = eig(M);
end

When applied to our hard example from before, RandDiag succeeds at giving a matrix that diagonalizes :

>> Q = rand_diag(C);
>> Q'*C*Q
ans =
1.0000 - 1.0000i  -0.0000 + 0.0000i
-0.0000 - 0.0000i   1.0000 + 1.0000i

For computing the matrix of eigenvectors for a unitary matrix, RandDiag takes 0.4 seconds, just as fast as the Hermitian eigendecomposition did.

>> tic; V_U = rand_diag(U); toc
Elapsed time is 0.437309 seconds.

He and Kressner’s algorithm is delightful. Ultimately, it uses randomness in only a small way. For most coefficients , a matrix diagonalizing will also diagonalize . But, for any specific choice of , there is a possibility of failure. To avoid this possibility, we can just pick and at random. It’s really as simple as that.

References: RandDiag was proposed in A simple, randomized algorithm for diagonalizing normal matrices by He and Kressner (2024), building on their earlier work in Randomized Joint Diagonalization of Symmetric Matrices (2022) which considers the general case of using random linear combinations to (approximately) simultaneous diagonalize (nearly) commuting matrices. RandDiag is an example of a linear algebraic algorithm that uses randomness to put the input into “general position”; see Randomized matrix computations: Themes and variations by Kireeva and Tropp (2024) for a discussion of this, and other, ways of using randomness to design matrix algorithms.

# Neat Randomized Algorithms: Randomized Cholesky QR

As a research area, randomized numerical linear algebra (RNLA) is as hot as ever. To celebrate the exciting work in this space, I’m starting a new series on my blog where I celebrate cool recent algorithms in the area. At some future point, I might talk about my own work in this series, but for now I’m hoping to use this series to highlight some of the awesome work being done by my colleagues.

Given a tall matrix with , its (economy-size) QR factorization is a decomposition of the form , where is a matrix with orthonormal columns and is upper triangular. QR factorizations are used to solve least-squares problems and as a computational procedure to orthonormalize the columns of a matrix.

Here’s an example in MATLAB, where we use QR factorization to orthonormalize the columns of a test matrix. It takes about 2.5 seconds to run.

>> A = randn(1e6, 1e2) * randn(1e2) * randn(1e2); % test matrix
>> tic; [Q,R] = qr(A,"econ"); toc
Elapsed time is 2.647317 seconds.

The classical algorithm for computing a QR factorization uses Householder reflectors and is exceptionally numerically stable. Since has orthonormal columns, is the identity matrix. Indeed, this relation holds up to a tiny error for the computed by Householder QR:

>> norm(Q'*Q - eye(1e2)) % || Q^T Q - I ||
ans =
7.0396e-14

The relative error is also small:

>> norm(A - Q*R) / norm(A)
ans =
4.8981e-14

Here is an alternate procedure for computing a QR factorization, known as Cholesky QR:

function [Q,R] = cholesky_qr(A)
R = chol(A'*A);
Q = A / R;       % Q = A * R^{-1}
end

This algorithm works by forming , computing its (upper triangular) Cholesky decomposition , and setting . Cholesky QR is very fast, about faster than Householder QR for this example:

>> tic; [Q,R] = cholesky_qr(A); toc
Elapsed time is 0.536694 seconds.

Unfortunately, Cholesky QR is much less accurate and numerically stable than Householder QR. Here, for instance, is the value of , about ten million times larger than for Householder QR!:

>> norm(Q'*Q - eye(1e2))
ans =
7.5929e-07

What’s going on? As we’ve discussed before on this blog, forming is typically problematic in linear algebraic computations. The “badness” of a matrix is measured by its condition number, defined to be the ratio of its largest and smallest singular values . The condition number of is the square of the condition number of , , which is at the root of Cholesky QR’s loss of accuracy. Thus, Cholesky QR is only appropriate for matrices that are well-conditioned, having a small condition number , say .

The idea of randomized Cholesky QR is to use randomness to precondition , producing a matrix that is well-conditioned. Then, since is well-conditioned, we can apply ordinary Cholesky QR to it without issue. Here are the steps:

1. Draw a sketching matrix of size ; see these posts of mine for an introduction to sketching.
2. Form the sketch . This step compresses the very tall matrix to the much shorter matrix of size .
3. Compute a QR factorization using Householder QR. Since the matrix is small, this factorization will be quick to compute.
4. Form the preconditioned matrix .
5. Apply Cholesky QR to to compute .
6. Set . Observe that , as desired.

MATLAB code for randomized Cholesky QR is provided below:1Code for the sparsesign subroutine can be found here.

function [Q,R] = rand_cholesky_qr(A)
S = sparsesign(2*size(A,2),size(A,1),8); % sparse sign embedding
R1 = qr(S*A,"econ"); % sketch and (Householder) QR factorize
B = A / R1; % B = A * R_1^{-1}
[Q,R2] = cholesky_qr(B);
R = R2*R1;
end

Randomized Cholesky QR is still faster than ordinary Householder QR, about faster in our experiment:

>> tic; [Q,R] = rand_cholesky_qr(A); toc
Elapsed time is 0.920787 seconds.

Randomized Cholesky QR greatly improves on ordinary Cholesky QR in terms of accuracy and numerical stability. In fact, the size of is even smaller than for Householder QR!

>> norm(Q'*Q - eye(1e2))
ans =
1.0926e-14

The relative error is small, too! Even smaller than for Householder QR in fact:

>> norm(A - Q*R) / norm(A)
ans =
4.0007e-16

Like many great ideas, randomized Cholesky QR was developed independently by a number of research groups. A version of this algorithm was first introduced in 2021 by Fan, Guo, and Lin. Similar algorithms were investigated in 2022 and 2023 by Balabanov, Higgins et al., and Melnichenko et al. Check out Melnichenko et al.‘s paper in particular, which shows very impressive results for using randomized Cholesky QR to compute column pivoted QR factorizations.

References: Primary references are A Novel Randomized XR-Based Preconditioned CholeskyQR Algorithm by Fan, Guo, and Lin (2021); Randomized Cholesky QR factorizations by Balabanov (2022); Analysis of Randomized Householder-Cholesky QR Factorization with Multisketching by Higgins et al. (2023); CholeskyQR with Randomization and Pivoting for Tall Matrices (CQRRPT) by Melnichenko et al. (2023). The idea of using sketching to precondition tall matrices originates in the paper A fast randomized algorithm for overdetermined linear least-squares regression by Rokhlin and Tygert (2008).

# Don’t Use Gaussians in Stochastic Trace Estimation

Suppose we are interested in estimating the trace of an matrix that can be only accessed through matrix–vector products . The classical method for this purpose is the GirardHutchinson estimator

where the vectors are independent, identically distributed (iid) random vectors satisfying the isotropy condition

Examples of vectors satisfying this condition include

Stochastic trace estimation has a number of applications: log-determinant computations in machine learningpartition function calculations in statistical physicsgeneralized cross validation for smoothing splines, and triangle counting in large networks. Several improvements to the basic Girard–Hutchinson estimator have been developed recently. I am partial to XTrace, an improved trace estimator that I developed with my collaborators.

This post is addressed at the question:

Which distribution should be used for the test vectors for stochastic trace estimation?

Since the Girard–Hutchinson estimator is unbiased , the variance of is equal to the mean-square error. Thus, the lowest variance trace estimate is the most accurate. In my previous post on trace estimation, I discussed formulas for the variance of the Girard–Hutchinson estimator with different choices of test vectors. In that post, I stated the formulas for different choices of test vectors (Gaussian, random signs, sphere) and showed how those formulas could be proven.

In this post, I will take the opportunity to editorialize on which distribution to pick. The thesis of this post is as follows:

The sphere distribution is essentially always preferable to the Gaussian distribution for trace estimation.

To explain why, let’s focus on the case when is real and symmetric.1The same principles hold in the general case, but the variance formulas are more delicate to state. See my previous post for the formulas. Let be the eigenvalues of and define the eigenvalue mean

Then the variance of the Girard–Hutchinson estimator with Gaussian vectors is

For vectors drawn from the sphere, we have

The sphere distribution improves on the Gaussian distribution in two ways. First, the variance of is smaller than by a factor of . This improvement is quite minor. Second, and more importantly, is proportional to the sum of ‘s squared eigenvalues whereas is proportional to the sum of ‘s squared eigenvalues after having been shifted to be mean-zero!

The difference between Gaussian and sphere test vectors can be large. To see this, consider a matrix with eigenvalues uniformly distributed between and with a (Haar orthgonal) random matrix of eigenvectors. For simplicity, since the variance of all Girard–Hutchinson estimates is proportional to , we take . Below show the variance of Girard–Hutchinson estimator for different distributions for the test vector. We see that the sphere distribution leads to a trace estimate which has a variance 300× smaller than the Gaussian distribution. For this example, the sphere and random sign distributions are similar.

## Which Distribution Should You Use: Signs vs. Sphere

The main point of this post is to argue against using the Gaussian distribution. But which distribution should you use: Random signs? The sphere distribution? The answer, for most applications, is one of those two, but exactly which depends on the properties of the matrix .

The variance of the Girard–Hutchinson estimator with the random signs estimator is

Thus, depends on the size of the off-diagonal entries of ; does not depend on the diagonal of at all! For matrices with small off-diagonal entries (such as diagonally dominant matrices), the random signs distribution is often the best.

However, for other problems, the sphere distribution is preferable to random signs. The sphere distribution is rotation-invariant, so is independent of the eigenvectors of the (symmetric) matrix , depending only on ‘s eigenvalues. By contrast, the variance of the Girard–Hutchinson estimator with the random signs distribution can significantly depend on the eigenvectors of the matrix . For a given set of eigenvalues and the worst-case choice of eigenvectors, will always be smaller than . In fact, is the minimum variance distribution for Girard–Hutchinson trace estimation for a matrix with fixed eigenvalues and worst-case eigenvectors; see this section of my previous post for details.

In my experience, random signs and the sphere distribution are both perfectly adequate for trace estimation and either is a sensible default if you’re developing software. The Gaussian distribution on the other hand… don’t use it unless you have a good reason to.

# How Good Can Stochastic Trace Estimates Be?

I am excited to share that our paper XTrace: Making the most of every sample in stochastic trace estimation has been published in the SIAM Journal on Matrix Analysis and Applications. (See also our paper on arXiv.)

Spurred by this exciting news, I wanted to take the opportunity to share one of my favorite results in randomized numerical linear algebra: a “speed limit” result of Meyer, Musco, Musco, and Woodruff that establishes a fundamental limitation on how accurate any trace estimation algorithm can be.

Let’s back up. Given an unknown square matrix , the trace of , defined to be the sum of its diagonal entries

The catch? We assume that we can only access the matrix through matrix–vector products (affectionately known as “matvecs”): Given any vector , we have access to . Our goal is to form an estimate that is as accurate as possible while using as few matvecs as we can get away with.

To simplify things, let’s assume the matrix is symmetric and positive (semi)definite. The classical algorithm for trace estimation is due to Girard and Hutchinson, producing a probabilistic estimate with a small average (relative) error:

If one wants high accuracy, this algorithm is expensive. To achieve just a 1% error () requires roughly matvecs!

This state of affairs was greatly improved by Meyer, Musco, Musco, and Woodruff. Building upon previous work, they proposed the Hutch++ algorithm and proved it outputs an estimate satisfying the following bound:

(1)

Now, we only require roughly matvecs to achieve 1% error! Our algorithm, XTrace, satisfies the same error guarantee (1) as Hutch++. On certain problems, XTrace can be quite a bit more accurate than Hutch++.

## The MMMW Trace Estimation “Speed Limit”

Given the dramatic improvement of Hutch++ and XTrace over Girard–Hutchinson, it is natural to hope: Is there an algorithm that does even better than Hutch++ and XTrace? For instance, is there an algorithm satisfying an even slightly better error bound of the form

Unfortunately not. Hutch++ and XTrace are essentially as good as it gets.

Let’s add some fine print. Consider an algorithm for the trace estimation problem. Whenever the algorithm wants, it can present a vector and receive back . The algorithm is allowed to be adaptive: It can use the matvecs it has already collected to decide which vector to present next. We measure the cost of the algorithm in terms of the number of matvecs alone, and the algorithm knows nothing about the psd matrix other what it learns from matvecs.

One final stipulation:

Simple entries assumption. We assume that the entries of the vectors presented by the algorithm are real numbers between and with up to digits after the decimal place.

To get a feel for this simple entries assumption, suppose we set . Then would be an allowed input vector, but would not be (too many digits after the decimal place). Similarly, would not be valid because its entries exceed . The simple entries assumption is reasonable as we typically represent numbers on digital computers by storing a fixed number of digits of accuracy.1We typically represent numbers on digital computers by floating point numbers, which essentially represent numbers using scientific notation like . For this analysis of trace estimation, we use fixed point numbers like (no powers of ten allowed)!

With all these stipulations, we are ready to state the “speed limit” for trace estimation proved by Meyer, Musco, Musco, and Woodruff:

Informal theorem (Meyer, Musco, Musco, Woodruff). Under the assumptions above, there is no trace estimation algorithm producing an estimate satisfying

We will see a slightly sharper version of the theorem below, but this statement captures the essence of the result.

## Communication Complexity

To prove the MMMW theorem, we have to take a journey to the beautiful subject of communication complexity. The story is this. Alice and Bob are interested in solving a computational problem together. Alice has her input and Bob has his input , and they are interested in computing a function of both their inputs.

Unfortunately for the two of them, Alice and Bob are separated by a great distance, and can only communicate by sending single bits (0 or 1) of information over a slow network connection. Every bit of communication is costly. The field of communication complexity is dedicated to determining how efficiently Alice and Bob are able to solve problems of this form.

The Gap-Hamming problem is one example of a problem studied in communication complexity. As inputs, Alice and Bob receive vectors with and entries from a third party Eve. Eve promises Alice and Bob that their vectors and satisfy one of two conditions:

(2)

Alice and Bob must work together, sending as few bits of communication as possible, to determine which case they are in.

There’s one simple solution to this problem: First, Bob sends his whole input vector to Alice. Each entry of takes one of the two value and can therefore be communicated in a single bit. Having received , Alice computes , determines whether they are in case 0 or case 1, and sends Bob a single bit to communicate the answer. This procedure requires bits of communication.

Can Alice and Bob still solve this problem with many fewer than bits of communication, say bits? Unfortunately not. The following theorem of Chakrabati and Regev shows that roughly bits of communication are needed to solve this problem:

Theorem (Chakrabati–Regev). Any algorithm which solves the Gap-Hamming problem that succeeds with at least probability for every pair of inputs and (satisfying one of the conditions (2)) must take bits of communication.

Here, is big-Omega notation, closely related to big-O notation and big-Theta notation . For the less familiar, it can be helpful to interpret , , and as all standing for “proportional to ”. In plain language, the theorem of Chakrabati and Regev result states that there is no algorithm for the Gap-Hamming problem that much more effective than the basic algorithm where Bob sends his whole input to Alice (in the sense of requiring less than bits of communication).

## Reducing Gap-Hamming to Trace Estimation

This whole state of affairs is very sad for Alice and Bob, but what does it have to do with trace estimation? Remarkably, we can use hardness of the Gap-Hamming problem to show there’s no algorithm that fundamentally improves on Hutch++ and XTrace. The argument goes something like this:

1. If there were a trace estimation algorithm fundamentally better than Hutch++ and XTrace, we could use it to solve Gap-Hamming in fewer than bits of communication.
2. But no algorithm can solve Gap-Hamming in fewer than bits or communication.
3. Therefore, no trace estimation algorithm is fundamentally better than Hutch++ and XTrace.

Step 2 is the work of Chakrabati and Regev, and step 3 follows logically from 1 and 2. Therefore, we are left to complete step 1 of the argument.

### Protocol

Assume we have access to a really good trace estimation algorithm. We will use it to solve the Gap-Hamming problem. For simplicity, assume is a perfect square. The basic idea is this:

• Have Alice and Bob reshape their inputs into matrices , and consider (but do not form!) the positive semidefinite matrix

• Observe that

Thus, the two cases in (2) can be equivalently written in terms of :

(2′)

• By working together, Alice and Bob can implement a trace estimation algorithm. Alice will be in charge of running the algorithm, but Alice and Bob must work together to compute matvecs. (Details below!)
• Using the output of the trace estimation algorithm, Alice determines whether they are in case 0 or 1 (i.e., where or ) and sends the result to Bob.

To complete this procedure, we just need to show how Alice and Bob can implement the matvec procedure using minimal communication. Suppose Alice and Bob want to compute for some vector with entries between and with up to decimal digits. First, convert to a vector whose entries are integers between and . Since , interconverting between and is trivial. Alice and Bob’s procedure for computing is as follows:

• Alice sends Bob .
• Having received , Bob forms and sends it to Alice.
• Having received , Alice computes and sends it to Bob.
• Having received , Bob computes and sends its to Alice.
• Alice forms .

Because and are and have entries, all vectors computed in this procedure are vectors of length with integer entries between and . We conclude the communication cost for one matvec is bits.

### Analysis

Consider an algorithm we’ll call BestTraceAlgorithm. Given any accuracy parameter , BestTraceAlgorithm requires at most matvecs and, for any positive semidefinite input matrix of any size, produces an estimate satisfying

(3)

We assume that BestTraceAlgorithm is the best possible algorithm in the sense that no algorithm can achieve (3) on all (positive semidefinite) inputs with matvecs.

To solve the Gap-Hamming problem, Alice and Bob just need enough accuracy in their trace estimation to distinguish between cases 0 and 1. In particular, if

then Alice and Bob can distinguish between cases 0 and 1 in (2′)

Suppose that Alice and Bob apply trace estimation to solve the Gap-Hamming problem, using matvecs in total. The total communication is bits. Chakrabati and Regev showed that Gap-Hamming requires bits of communication (for some ) to solve the Gap-Hamming problem with probability. Thus, if , then Alice and Bob fail to solve the Gap-Hamming problem with at least probability. Thus,

The contrapositive of this statement is that if

Say Alice and Bob run BestTraceAlgorithm with parameter . Then, by (3) and Markov’s inequality,

Therefore, BestTraceAlgorithm requires at least

Using the fact that we’ve set , we conclude that any trace estimation algorithm, even BestTraceAlgorithm, requires

In particular, no trace estimation algorithm can achieve mean relative error using even matvecs. This proves the MMMW theorem.

# Five Interpretations of Kernel Quadrature

I’m excited to share that my paper Kernel quadrature with randomly pivoted Cholesky, joint with Elvira Moreno, has been accepted to NeurIPS 2023 as a spotlight.

Today, I want to share with you a little about the kernel quadrature problem. To avoid this post getting too long, I’m going to write this post assuming familiarity with the concepts of reproducing kernel Hilbert spaces and Gaussian processes.

Integration is one of the most widely used operations in mathematics and its applications. As such, it is a basic problem of wide interest to develop numerical methods for evaluating integrals.

In this post, we will consider a quite general integration problem. Let be a domain and let be a (finite Borel) measure on . We consider the task of evaluating

One can imagine that , , and are fixed, but we may want to evaluate this same integral for multiple different functions .

To evaluate, we will design a quadrature approximation to the integral :

Concretely, we wish to find real numbers and points such that the approximation is accurate.

## Smoothness and Reproducing Kernel Hilbert Spaces

As is frequently the case in computational mathematics, the accuracy we can expect for this integration problem depends on the smoothness of the integrand . The more smooth is, the more accurately we can expect to compute for a given budget of computational effort.

In this post, will measure smoothness using the reproducing kernel Hilbert space (RKHS) formalism. Let be an RKHS with norm . We can interpret the norm as assigning a roughness to each function . If is large, then is rough; if is small, then is smooth.

Associated to the RKHS is the titular reproducing kernel . The kernel is a bivariate function . It is related to the RKHS inner product by the reproducing property

Here, represents the univariate function obtained by setting the first input of to be .

## The Ideal Weights

To design a quadrature rule, we have to set the nodes and weights . Let’s first assume that the nodes are fixed, and talk about how to pick the weights .

There’s one choice of weights that we’ll called the ideal weights. There (at least) are five equivalent ways of characterizing the ideal weights. We’ll present all of them. As an exercise, you can try and convince yourself that these characterizations are equivalent, giving rise to the same weights.

### Interpretation 1: Exactness

A standard way of designing quadrature rules is to make them exact (i.e., error-free) for some class of functions. For instance, many classical quadrature rules are exact for polynomials of degree up to .

For kernel quadrature, it makes sense to design the quadrature rule to be exact for the kernel function at the selected nodes. That is, we require

Enforcing exactness gives us linear equations for the unknowns :

Under mild conditions, this system of linear equations is uniquely solvable, and the solution is the ideal weights.

### Interpretation 2: Interpolate and Integrate

Here’s another very classical way of designing a quadrature rule. First, interpolate the function values at the nodes, obtaining an interpolant . Then, obtain an approximation to the integral by integrating the interpolant:

In our context, the appropriate interpolation method is kernel interpolation.1Kernel interpolation is also called Gaussian process regression or kriging though (confusingly) these terms can also refer to slightly different methods. It is the regularization-free limit of kernel ridge regression. The kernel interpolant is defined to be the minimum-norm function that interpolates the data:

Remarkably, this infinite-dimensional problem has a tractably computable solution. In fact, is the unique function of the form

that agrees with on the points .With a little algebra, you can show that the integral of is

where are the ideal weights.

### Interpretation 3: Minimizing the Worst-Case Error

Define the worst-case error of weights and nodes to be

The quantity is the highest possible quadrature error for a function of norm at most 1.

Having defined the worst-case error, the ideal weights are precisely the weights that minimize this quantity

### Interpretation 4: Minimizing the Mean-Square Error

The next two interpretations of the ideal weights will adopt a probabilistic framing. A Gaussian process is a random function such that ’s values at any collection of points are (jointly) Gaussian random variables. We write for a mean-zero Gaussian process with covariance function :

Define the mean-square quadrature error of weights and nodes to be

The mean-square error reports the expected squared quadrature error over all functions drawn from a Gaussian process with covariance function .

Pleasantly, the mean-square error is equal ro the square of the worst-case error

As such, the ideal weights also minimize the mean-square error

### Interpretation 5: Conditional Expectation

For our last interpretation, again consider a Gaussian process . The integral of this random function is a random variable. To numerically integrate a function , compute the expectation of conditional on agreeing with at the quadrature nodes:

One can show that this procedure yields the quadrature scheme with the ideal weights.

### Conclusion

We’ve just seen five sensible ways of defining the ideal weights for quadrature in a general reproducing kernel Hilbert space. Remarkably, all five lead to exactly the same choice of weights. To me, these five equivalent characterizations give me more confidence that the ideal weights really are the “right” or “natural” choice for kernel quadrature.

That said, there are other reasonable requirements that we might want to impose on the weights. For instance, if is a probability measure and , it is reasonable to add an additional constraint that the weights lie in the probability simplex

With this additional stipulation, a quadrature rule can be interpreted as integrating against a discrete probability measure ; thus, in effect, quadrature amounts to approximating one probability measure by another . Additional constraints such as these can easily be imposed when using the optimization characterizations 3 and 4 of the ideal weights. See this paper for details.

We’ve spent a lot of time talking about how to pick the quadrature weights, but how should we pick the nodes ? To pick the nodes, it seems sensible to try and minimize the worst-case error with the ideal weights . For this purpose, we can use the following formula:

Here, is the Nyström approximation to the kernel induced by the nodes , defined to be

We have written for the kernel matrix with entry and and for the row and column vectors with th entry and .

I find the appearance of the Nyström approximation in this context to be surprising and delightful. Previously on this blog, we’ve seen (column) Nyström approximation in the context of matrix low-rank approximation. Now, a continuum analog of the matrix Nyström approximation has appeared in the error formula for numerical integration.

The appearance of the Nyström approximation in the kernel quadrature error also suggests a strategy for picking the nodes.

Node selection strategy. We should pick the nodes to make the Nyström approximation as accurate as possible.

The closer is to , the smaller the function is and, thus, the smaller the error

Fortunately, we have randomized matrix algorithms for picking good nodes for matrix Nyström approximation such as randomly pivoted Cholesky, ridge leverage score sampling, and determinantal point process sampling; maybe these matrix tools can be ported to the continuous kernel setting?

Indeed, all three of these algorithms—randomly pivoted Cholesky, ridge leverage score sampling, and determinantal point process sampling—have been studied for kernel quadrature. The first of these algorithms, randomly pivoted Cholesky, is the subject of our paper. We show that this simple, adaptive sampling method produces excellent nodes for kernel quadrature. Intuitively, randomly pivoted Cholesky is effective because it is repulsive: After having picked nodes , it has a high probability of placing the next node far from the previously selected nodes.

The following image shows 20 nodes selected by randomly pivoted Cholesky in a crescent-shaped region. The cyan–pink shading denotes the probability distribution for picking the next node. We see that the center of the crescent does not have any nodes, and thus is most likely to receive a node during the next round of sampling.

In our paper, we demonstrate—empirically and theoretically—that randomly pivoted Cholesky produces excellent nodes for quadrature. We also discuss efficient rejection sampling algorithms for sampling nodes with the randomly pivoted Cholesky distribution. Check out the paper for details!

# Which Sketch Should I Use?

This is the second of a sequence of two posts on sketching, which I’m doing on the occasion of my new paper on the numerical stability of the iterative sketching method. For more on what sketching is and how it can be used to solve computational problems, I encourage you to check out the first post.

The goals of this post are more narrow. I seek to answer the question:

Which sketching matrix should I use?

To cut to the chase, my answer to this question is:

Sparse sign embeddings are a sensible default for sketching.

There are certainly cases when sparse sign embeddings are not the best type of sketch to use, but I hope to convince you that they’re a good sketching matrix to use for most purposes.

## Experiments

Let’s start things off with some numerical experiments.1Code for all numerical experiments can be found on the blogpost branch of the Github for my recent paper. We’ll compare three types of sketching matrices: Gaussian embeddings, a subsampled randomized trigonometric transform (SRTT), and sparse sign embeddings. See the last post for descriptions of these sketching matrices. I’ll discuss a few additional types of sketching matrices that require more discussion at the end of this post.

Recall that a sketching matrix seeks to compress a high-dimensional matrix or vector to a lower-dimensional sketched matrix or vector . The quality of a sketching matrix for a matrix is measured by its distortion , defined to be the smallest number such that

Here, denotes the column space of the matrix .

### Timing

We begin with timing test. We will measure three different times for each embedding:

1. Construction. The time required to generate the sketching matrix .
2. Vector apply. The time to apply the sketch to a single vector.
3. Matrix apply. The time to apply the sketch to an matrix.

We will test with input dimension (one million) and output dimension . For the SRTT, we use the discrete cosine transform as our trigonometric transform. For the sparse sign embedding, we use a sparsity parameter .

Here are the results (timings averaged over 20 trials):

Our conclusions are as follows:

• Sparse sign embeddings are definitively the fastest to apply, being 3–20× faster than the SRTT and 74–100× faster than Gaussian embeddings.
• Sparse sign embeddings are modestly slower to construct than the SRTT, but much faster to construct than Gaussian embeddings.

Overall, the conclusion is that sparse sign embeddings are the fastest sketching matrices by a wide margin: For an “end-to-end” workflow involving generating the sketching matrix and applying it to a matrix , sparse sign embeddings are 14× faster than SRTTs and 73× faster than Gaussian embeddings.2More timings are reported in Table 1 of this paper, which I credit for inspiring my enthusiasm for the sparse sign embedding l.

### Distortion

Runtime is only one measure of the quality of a sketching matrix; we also must care about the distortion . Fortunately, for practical purposes, Gaussian embeddings, SRTTs, and sparse sign embeddings all tend to have similar distortions. Therefore, we are free to use the sparse sign embeddings, as they as typically are the fastest.

Consider the following test. We generate a sparse random test matrix of size for and using the MATLAB sprand function; we set the sparsity level to 1%. We then compare the distortion of Gaussian embeddings, SRTTs, and sparse sign embeddings across a range of sketching dimensions between 100 and 10,000. We report the distortion averaged over 100 trials. The theoretically predicted value (equivalently, ) is shown as a dashed line.

To me, I find these results remarkable. All three embeddings exhibit essentially the same distortion parameter predicted by the Gaussian theory.

It would be premature to declare success having only tested on one type of test matrix . Consider the following four test matrices:

• Sparse: The test matrix from above.
• Dense: is taken to be a matrix with independent standard Gaussian random values.
• Khatri–Rao: is taken to be the Khatri–Rao product of three Haar random orthogonal matrices.
• Identity: is taken to be the identity matrix stacked onto a matrix of zeros.

The performance of sparse sign embeddings (again with sparsity parameter ) is shown below:

We see that for the first three test matrices, the performance closely follows the expected value . However, for the last test matrix “Identity”, we see the distortion begins to slightly exceed this predicted distortion for .

To improve sparse sign embeddings for higher values of , we can increase the value of the sparsity parameter . We recommend

With this higher sparsity level, the distortion closely tracks for all four test matrices:

### Conclusion

Implemented appropriately (see below), sparse sign embeddings can be faster than other sketching matrices by a wide margin. The parameter choice is enough to ensure the distortion closely tracks for most test matrices. For the toughest test matrices, a slightly larger sparsity parameter can be necessary to achieve the optimal distortion.

While these tests are far from comprehensive, they are consistent with the uniformly positive results for sparse sign embeddings reported in the literature. We believe that this evidence supports the argument that sparse sign embeddings are a sensible default sketching matrix for most purposes.

## Sparse Sign Embeddings: Theory and Practice

Given the highly appealing performance characteristics of sparse sign embeddings, it is worth saying a few more words about these embeddings and how they perform in both theory and practice.

Recall that a sparse sign embedding is a random matrix of the form

Each column is an independent and randomly generated to contain exactly nonzero entries in uniformly random positions. The value of each nonzero entry of is chosen to be either or with 50/50 odds.

### Parameter Choices

The goal of sketching is to reduce vectors of length to a smaller dimension . For linear algebra applications, we typically want to preserve all vectors in the column space of a matrix up to distortion :

To use sparse sign embeddings, we must choose the parameters appropriately:

Given a dimension and a target distortion , how do we pick and ?

Based on the experiments above (and other testing reported in the literature), we recommend the following parameter choices in practice:

The parameter choice is advocated by Tropp, Yurtever, Udell, and Cevher; they mention experimenting with parameter values as small as . The value has demonstrated deficiencies and should almost always be avoided (see below). The scaling is derived from the analysis of Gaussian embeddings. As Martinsson and Tropp argue, the analysis of Gaussian embeddings tends to be reasonably descriptive of other well-designed random embeddings.

The best-known theoretical analysis, due to Cohen, suggests more cautious parameter setting for sparse sign embeddings:

The main difference between Cohen’s analysis and the parameter recommendations above is the presence of the factor and the lack of explicit constants in the O-notation.

### Implementation

For good performance, it is imperative to store using either a purpose-built data structure or a sparse matrix format (such as a MATLAB sparse matrix or scipy sparse array).

If a sparse matrix library is unavailable, then either pursue a dedicated implementation or use a different type of embedding; sparse sign embeddings are just as slow as Gaussian embeddings if they are stored in an ordinary non-sparse matrix format!

Even with a sparse matrix format, it can require care to generate and populate the random entries of the matrix . Here, for instance, is a simple function for generating a sparse sign matrix in MATLAB:

function S = sparsesign_slow(d,n,zeta)
cols = kron((1:n)',ones(zeta,1)); % zeta nonzeros per column
vals = 2*randi(2,n*zeta,1) - 3; % uniform random +/-1 values
rows = zeros(n*zeta,1);
for i = 1:n
rows((i-1)*zeta+1:i*zeta) = randsample(d,zeta);
end
S = sparse(rows, cols, vals / sqrt(zeta), d, n);
end

Here, we specify the rows, columns, and values of the nonzero entries before assembling them into a sparse matrix using the MATLAB sparse command. Since there are exactly nonzeros per column, the column indices are easy to generate. The values are uniformly and can also be generated using a single line. The real challenge to generating sparse sign embeddings in MATLAB is the row indices, since each batch of row indices much be chosen uniformly at random between and without replacement. This is accomplished in the above code by a for loop, generating row indices at a time using the slow randsample function.

As its name suggests, the sparsesign_slow is very slow. To generate a sparse sign embedding with sparsity requires 53 seconds!

Fortunately, we can do (much) better. By rewriting the code in C and directly generating the sparse matrix in the CSC format MATLAB uses, generating this same 200 by 10 million sparse sign embedding takes just 0.4 seconds, a speedup of 130× over the slow MATLAB code. A C implementation of the sparse sign embedding that can be used in MATLAB using the MEX interface can be found in this file in the Github repo for my recent paper.

## Other Sketching Matrices

Let’s leave off the discussion by mentioning other types of sketching matrices not considered in the empirical comparison above.

### Coordinate Sampling

Another family of sketching matrices that we haven’t talked about are coordinate sampling sketches. A coordinate sampling sketch consists of indices and weights . To apply , we sample the indices and reweight them using the weights:

Coordinate sampling is very appealing: To apply to a matrix or vector requires no matrix multiplication of trigonometric transforms, just picking out some entries or rows and rescaling them.

In order for coordinate sampling to be effective, we need to pick the right indices. Below, we compare two coordinate sampling sketching approaches, uniform sampling and leverage score sampling (both with replacement), to the sparse sign embedding with the suggested parameter setting for the hard “Identity” test matrix used above.

We see right away that the uniform sampling fails dramatically on this problem. That’s to be expected. All but 50 of 100,000 rows of are zero, so picking rows uniformly at random will give nonsense with very high probability. Uniform sampling can work well for matrices which are “incoherent”, with all rows of being of “similar importance”.

Conclusion (Uniform sampling). Uniform sampling is a risky method; it works excellently for some problems, but fails spectacularly for others. Use only with caution!

The ridge leverage score sampling method is more interesting. Unlike all the other sketches we’ve discussed in this post, ridge leverage score sampling is data-dependent. First, it computes a leverage score for each row of and then samples rows with probabilities proportional to these scores. For high enough values of , ridge leverage score sampling performs slightly (but only slightly) worse than the characteristic scaling we expect for an oblivious subspace embedding.

Ultimately, leverage score sampling has two disadvantages when compared with oblivious sketching matrices:

• Higher distortion, higher variance. The distortion of a leverage score sketch is higher on average, and more variable, than an oblivious sketch, which achieve very consistent performance.
• Computing the leverage scores. In order to implement this sketch, the leverage scores have to first be computed or estimated. This is a nontrivial algorithmic problem; the most direct way of computing the leverage scores requires a QR decomposition at cost, much higher than other types of sketches.

There are settings when coordinate sampling methods, such as leverage scores, are well-justified:

• Structured matrices. For some matrices , the leverage scores might be very cheap to compute or approximate. In such cases, coordinate sampling can be faster than oblivious sketching.
• “Active learning”. For some problems, each entry of the vector or row of the matrix may be expensive to generate. In this case, coordinate sampling has the distinct advantage that computing or only requires generating the entries of or rows of for the randomly selected indices .

Ultimately, oblivious sketching and coordinate sampling both have their place as tools in the computational toolkit. For the reasons described above, I believe that oblivious sketching should usually be preferred to coordinate sampling in the absence of a special reason to prefer the latter.

### Tensor Random Embeddings

There are a number of sketching matrices with tensor structure; see here for a survey. These types of sketching matrices are very well-suited to tensor computations. If tensor structure is present in your application, I would put these types of sketches at the top of my list for consideration.

### CountSketch

The CountSketch sketching matrix is the case of the sparse sign embedding. CountSketch has serious deficiencies, and should only be used in practice with extreme care.

Consider the “Identity” test matrix from above but with parameter , and compare the distortion of CountSketch to the sparse sign embedding with parameters :

We see that the distortion of the CountSketch remains persistently high at 100% until the sketching dimension is taken , 20× higher than .

CountSketch is bad because it requires to be proportional to in order to achieve distortion . For all of the other sketching matrices we’ve considered, we’ve only required to be proportional to (or perhaps ). This difference between for CountSketch and for other sketching matrices is a at the root of CountSketch’s woefully bad performance on some inputs.3Here, the symbol is an informal symbol meaning “proportional to”.

The fact that CountSketch requires is simple to show. It’s basically a variant on the famous birthday problem. We include a discussion at the end of this post.4In fact, any oblivious sketching matrix with only 1 nonzero entry per column must have . This is Theorem 16 in the following paper.

There are ways of fixing the CountSketch. For instance, we can use a composite sketch , where is a CountSketch of size and is a Gaussian sketching matrix of size .5This construction is from this paper. For most applications, however, salvaging CountSketch doesn’t seem worth it; sparse sign embeddings with even nonzeros per column are already way more effective and reliable than a plain CountSketch.

## Conclusion

By now, sketching is quite a big field, with dozens of different proposed constructions for sketching matrices. So which should you use? For most use cases, sparse sign embeddings are a good choice; they are fast to construct and apply and offer uniformly good distortion across a range of matrices.

Bonus: CountSketch and the Birthday Problem
The point of this bonus section is to prove the following (informal) theorem:

Let be the “Identity” test matrix above. If is a CountSketch matrix with output dimension , then the distortion of for is with high probability.

Let’s see why. By the structure of the matrix , has the form

where each vector has a single in a uniformly random location .

Suppose that the indices are not all different from each other, say . Set , where is the standard basis vector with in position and zeros elsewhere. Then, but . Thus, for the distortion relation

to hold, . Thus,

For a moment, let’s put aside our analysis of the CountSketch, and turn our attention to a famous puzzle, known as the birthday problem:

How many people have to be in a room before there’s at least a 50% chance that two people share the same birthday?

The counterintuitive or “paradoxical” answer: 23. This is much smaller than many people’s intuition, as there are 365 possible birthdays and 23 is much smaller than 365.

The reason for this surprising result is that, in a room of 23 people, there are pairs of people. Each pair of people has a chance of sharing a birthday, so the expected number of birthdays in a room of 23 people is . Since are 0.69 birthdays shared on average in a room of 23 people, it is perhaps less surprising that 23 is the critical number at which the chance of two people sharing a birthday exceeds 50%.

Hopefully, the similarity between the birthday problem and CountSketch is becoming clear. Each pair of indices and in CountSketch have a chance of being the same. There are pairs of indices, so the expected number of equal indices is . Thus, we should anticipate is required to ensure that are distinct with high probability.

Let’s calculate things out a bit more precisely. First, realize that

To compute the probability that are distinct, imagine introducing each one at a time. Assuming that are all distinct, the probability are distinct is just the probability that does not take any of the values . This probability is

Thus, by the chain rule for probability,

To bound this quantity, use the numeric inequality for every , obtaining

Thus, we conclude that

Solving this inequality, we conclude that

This is a quantitative version of our informal theorem from earlier.

# Does Sketching Work?

I’m excited to share that my paper, Fast and forward stable randomized algorithms for linear least-squares problems has been released as a preprint on arXiv.

With the release of this paper, now seemed like a great time to discuss a topic I’ve been wanting to write about for a while: sketching. For the past two decades, sketching has become a widely used algorithmic tool in matrix computations. Despite this long history, questions still seem to be lingering about whether sketching really works:

In this post, I want to take a critical look at the question “does sketching work”? Answering this question requires answering two basic questions:

1. What is sketching?
2. What would it mean for sketching to work?

I think a large part of the disagreement over the efficacy of sketching boils down to different answers to these questions. By considering different possible answers to these questions, I hope to provide a balanced perspective on the utility of sketching as an algorithmic primitive for solving linear algebra problems.

## Sketching

In matrix computations, sketching is really a synonym for (linear) dimensionality reduction. Suppose we are solving a problem involving one or more high-dimensional vectors or perhaps a tall matrix . A sketching matrix is a matrix where . When multiplied into a high-dimensional vector or tall matrix , the sketching matrix produces compressed or “sketched” versions and that are much smaller than the original vector and matrix .

Let be a collection of vectors. For to be a “good” sketching matrix for , we require that preserves the lengths of every vector in up to a distortion parameter :

(1)

for every in .

For linear algebra problems, we often want to sketch a matrix . In this case, the appropriate set that we want our sketch to be “good” for is the column space of the matrix , defined to be

Remarkably, there exist many sketching matrices that achieve distortion for with an output dimension of roughly . In particular, the sketching dimension is proportional to the number of columns of . This is pretty neat! We can design a single sketching matrix which preserves the lengths of all infinitely-many vectors in the column space of .

## Sketching Matrices

There are many types of sketching matrices, each with different benefits and drawbacks. Many sketching matrices are based on randomized constructions in the sense that entries of are chosen to be random numbers. Broadly, sketching matrices can be classified into two types:

• Data-dependent sketches. The sketching matrix is constructed to work for a specific set of input vectors .
• Oblivious sketches. The sketching matrix is designed to work for an arbitrary set of input vectors of a given size (i.e., has elements) or dimension ( is a -dimensional linear subspace).

We will only discuss oblivious sketching for this post. We will look at three types of sketching matrices: Gaussian embeddings, subsampled randomized trignometric transforms, and sparse sign embeddings.

The details of how these sketching matrices are built and their strengths and weaknesses can be a little bit technical. All three constructions are independent from the rest of this article and can be skipped on a first reading. The main point is that good sketching matrices exist and are fast to apply: Reducing to requires roughly operations, rather than the operations we would expect to multiply a matrix and a vector of length .1Here, is big O notation.

### Gaussian Embeddings

The simplest type of sketching matrix is obtained by (independently) setting every entry of to be a Gaussian random number with mean zero and variance . Such a sketching matrix is called a Gaussian embedding.2Here, embedding is a synonym for sketching matrix.

Benefits. Gaussian embeddings are simple to code up, requiring only a standard matrix product to apply to a vector or matrix . Gaussian embeddings admit a clean theoretical analysis, and their mathematical properties are well-understood.

Drawbacks. Computing for a Gaussian embedding costs operations, significantly slower than the other sketching matrices we will consider below. Additionally, generating and storing a Gaussian embedding can be computationally expensive.

### Subsampled Randomized Trigonometric Transforms

The subsampled randomized trigonometric transform (SRTT) sketching matrix takes a more complicated form. The sketching matrix is defined to be a scaled product of three matrices

These matrices have the following definitions:

• is a diagonal matrix whose entries are each a random (chosen independently with equal probability).
• is a fast trigonometric transform such as a fast discrete cosine transform.3One can also use the ordinary fast Fourier transform, but this results in a complex-valued sketch.
• is a selection matrix. To generate , let be a random subset of , selected without replacement. is defined to be a matrix for which for every vector .

To store on a computer, it is sufficient to store the diagonal entries of and the selected coordinates defining . Multiplication of against a vector should be carried out by applying each of the matrices , , and in sequence, such as in the following MATLAB code:

% Generate randomness for S
signs = 2*randi(2,m,1)-3; % diagonal entries of D
idx = randsample(m,d); % indices i_1,...,i_d defining R

% Multiply S against b
c = signs .* b % multiply by D
c = dct(c) % multiply by F
c = c(idx) % multiply by R
c = sqrt(n/d) * c % scale

Benefits. can be applied to a vector in operations, a significant improvement over the cost of a Gaussian embedding. The SRTT has the lowest memory and random number generation requirements of any of the three sketches we discuss in this post.

Drawbacks. Applying to a vector requires a good implementation of a fast trigonometric transform. Even with a high-quality trig transform, SRTTs can be significantly slower than sparse sign embeddings (defined below).4For an example, see Figure 2 in this paper. SRTTs are hard to parallelize.5Block SRTTs are more parallelizable, however. In theory, the sketching dimension should be chosen to be , larger than for a Gaussian sketch.

### Sparse Sign Embeddings

A sparse sign embedding takes the form

Here, each column is an independently generated random vector with exactly nonzero entries with random values in uniformly random positions. The result is a matrix with only nonzero entries. The parameter is often set to a small constant like in practice.6This recommendation comes from the following paper, and I’ve used this parameter setting successfully in my own work.

Benefits. By using a dedicated sparse matrix library, can be very fast to apply to a vector (either or operations) to apply to a vector, depending on parameter choices (see below). With a good sparse matrix library, sparse sign embeddings are often the fastest sketching matrix by a wide margin.

Drawbacks. To be fast, sparse sign embeddings requires a good sparse matrix library. They require generating and storing roughly random numbers, higher than SRTTs (roughly numbers) but much less than Gaussian embeddings ( numbers). In theory, the sketching dimension should be chosen to be and the sparsity should be set to ; the theoretically sanctioned sketching dimension (at least according to existing theory) is larger than for a Gaussian sketch. In practice, we can often get away with using and .

### Summary

Using either SRTTs or sparse maps, a sketching a vector of length down to dimensions requires only to operations. To apply a sketch to an entire matrix thus requires roughly operations. Therefore, sketching offers the promise of speeding up linear algebraic computations involving , which typically take operations.

## How Can You Use Sketching?

The simplest way to use sketching is to first apply the sketch to dimensionality-reduce all of your data and then apply a standard algorithm to solve the problem using the reduced data. This approach to using sketching is called sketch-and-solve.

As an example, let’s apply sketch-and-solve to the least-squares problem:

(2)

We assume this problem is highly overdetermined with having many more rows than columns .

To solve this problem with sketch-and-solve, generate a good sketching matrix for the set . Applying to our data and , we get a dimensionality-reduced least-squares problem

(3)

The solution is the sketch-and-solve solution to the least-squares problem, which we can use as an approximate solution to the original least-squares problem.

Least-squares is just one example of the sketch-and-solve paradigm. We can also use sketching to accelerate other algorithms. For instance, we could apply sketch-and-solve to clustering. To cluster data points , first apply sketching to obtain and then apply an out-of-the-box clustering algorithms like k-means to the sketched data points.

## Does Sketching Work?

Most often, when sketching critics say “sketching doesn’t work”, what they mean is “sketch-and-solve doesn’t work”.

To address this question in a more concrete setting, let’s go back to the least-squares problem (2). Let denote the optimal least-squares solution and let be the sketch-and-solve solution (3). Then, using the distortion condition (1), one can show that

If we use a sketching matrix with a distortion of , then this bound tells us that

(4)

Is this a good result or a bad result? Ultimately, it depends. In some applications, the quality of a putative least-squares solution is can be assessed from the residual norm . For such applications, the bound (4) ensures that is at most twice . Often, this means is a pretty decent approximate solution to the least-squares problem.

For other problems, the appropriate measure of accuracy is the so-called forward error , measuring how close is to . For these cases, it is possible that might be large even though the residuals are comparable (4).

Let’s see an example, using the MATLAB code from my paper:

[A, b, x, r] = random_ls_problem(1e4, 1e2, 1e8, 1e-4); % Random LS problem
S = sparsesign(4e2, 1e4, 8); % Sparse sign embedding
sketch_and_solve = (S*A) \ (S*b); % Sketch-and-solve
direct = A \ b; % MATLAB mldivide

Here, we generate a random least-squares problem of size 10,000 by 100 (with condition number and residual norm ). Then, we generate a sparse sign embedding of dimension (corresponding to a distortion of roughly ). Then, we compute the sketch-and-solve solution and, as reference, a “direct” solution by MATLAB’s \.

We compare the quality of the sketch-and-solve solution to the direct solution, using both the residual and forward error:

fprintf('Residuals: sketch-and-solve %.2e, direct %.2e, optimal %.2e\n',...
norm(b-A*sketch_and_solve), norm(b-A*direct), norm(r))
fprintf('Forward errors: sketch-and-solve %.2e, direct %.2e\n',...
norm(x-sketch_and_solve), norm(x-direct))

Here’s the output:

Residuals: sketch-and-solve 1.13e-04, direct 1.00e-04, optimal 1.00e-04
Forward errors: sketch-and-solve 1.06e+03, direct 8.08e-07

The sketch-and-solve solution has a residual norm of , close to direct method’s residual norm of . However, the forward error of sketch-and-solve is nine orders of magnitude larger than the direct method’s forward error of .

Does sketch-and-solve work? Ultimately, it’s a question of what kind of accuracy you need for your application. If a small-enough residual is all that’s needed, then sketch-and-solve is perfectly adequate. If small forward error is needed, sketch-and-solve can be quite bad.

One way sketch-and-solve can be improved is by increasing the sketching dimension and lowering the distortion . Unfortunately, improving the distortion of the sketch is expensive. Because of the relation , to decrease the distortion by a factor of ten requires increasing the sketching dimension by a factor of one hundred! Thus, sketch-and-solve is really only appropriate when a low degree of distortion is necessary.

## Iterative Sketching: Combining Sketching with Iteration

Sketch-and-solve is a fast way to get a low-accuracy solution to a least-squares problem. But it’s not the only way to use sketching for least-squares. One can also use sketching to obtain high-accuracy solutions by combining sketching with an iterative method.

There are many iterative methods for least-square problems. Iterative methods generate a sequence of approximate solutions that we hope will converge at a rapid rate to the true least-squares solution, .

To using sketching to solve least-squares problems iteratively, we can use the following observation:

If is a sketching matrix for , then .

Therefore, if we compute a QR factorization

then

Notice that we used the fact that since has orthonormal columns. The conclusion is that .

Let’s use the approximation to solve the least-squares problem iteratively. Start off with the normal equations7As I’ve described in a previous post, it’s generally inadvisable to solve least-squares problems using the normal equations. Here, we’re just using the normal equations as a conceptual tool to derive an algorithm for solving the least-squares problem.

(5)

We can obtain an approximate solution to the least-squares problem by replacing by in (5) and solving. The resulting solution is

This solution will typically not be a good solution to the least-squares problem (2), so we need to iterate. To do so, we’ll try and solve for the error . To derive an equation for the error, subtract from both sides of the normal equations (5), yielding

Now, to solve for the error, substitute for again and solve for , obtaining a new approximate solution :

We can now go another step: Derive an equation for the error , approximate , and obtain a new approximate solution . Continuing this process, we obtain an iteration

(6)

This iteration is known as the iterative sketching method.8The name iterative sketching is for historical reasons. Original versions of the procedure involved taking a fresh sketch at every iteration. Later, it was realized that a single sketch suffices, albeit with a slower convergence rate. Typically, only having to sketch and QR factorize once is worth the slower convergence rate. If the distortion is small enough, this method converges at an exponential rate, yielding a high-accuracy least squares solution after a few iterations.

Let’s apply iterative sketching to the example we considered above. We show the forward error of the sketch-and-solve and direct methods as horizontal dashed purple and red lines. Iterative sketching begins at roughly the forward error of sketch-and-solve, with the error decreasing at an exponential rate until it reaches that of the direct method over the course of fourteen iterations. For this problem, at least, iterative sketching gives high-accuracy solutions to the least-squares problem!

To summarize, we’ve now seen two very different ways of using sketching:

• Sketch-and-solve. Sketch the data and and solve the sketched least-squares problem (3). The resulting solution is cheap to obtain, but may have low accuracy.
• Iterative sketching. Sketch the matrix and obtain an approximation to . Use the approximation to produce a sequence of better-and-better least-squares solutions by the iteration (6). If we run for enough iterations , the accuracy of the iterative sketching solution can be quite high.

By combining sketching with more sophisticated iterative methods such as conjugate gradient and LSQR, we can get an even faster-converging least-squares algorithm, known as sketch-and-precondition. Here’s the same plot from above with sketch-and-precondition added; we see that sketch-and-precondition converges even faster than iterative sketching does!

“Does sketching work?” Even for a simple problem like least-squares, the answer is complicated:

A direct use of sketching (i.e., sketch-and-solve) leads to a fast, low-accuracy solution to least-squares problems. But sketching can achieve much higher accuracy for least-squares problems by combining sketching with an iterative method (iterative sketching and sketch-and-precondition).

We’ve focused on least-squares problems in this section, but these conclusions could hold more generally. If “sketching doesn’t work” in your application, maybe it would if it was combined with an iterative method.

## Just How Accurate Can Sketching Be?

We left our discussion of sketching-plus-iterative-methods in the previous section on a positive note, but there is one last lingering question that remains to be answered. We stated that iterative sketching (and sketch-and-precondition) converge at an exponential rate. But our computers store numbers to only so much precision; in practice, the accuracy of an iterative method has to saturate at some point.

An (iterative) least-squares solver is said to be forward stable if, when run for a sufficient number of iterations, the final accuracy is comparable to accuracy of a standard direct method for the least-squares problem like MATLAB’s \ command or Python’s scipy.linalg.lstsq. Forward stability is not about speed or rate of convergence but about the maximum achievable accuracy.

The stability of sketch-and-precondition was studied in a recent paper by Meier, Nakatsukasa, Townsend, and Webb. They demonstrated that, with the initial iterate , sketch-and-precondition is not forward stable. The maximum achievable accuracy was worse than standard solvers by orders of magnitude! Maybe sketching doesn’t work after all?

Fortunately, there is good news:

• The iterative sketching method is provably forward stable. This result is shown in my newly released paper; check it out if you’re interested!
• If we use the sketch-and-solve method as the initial iterate for sketch-and-precondition, then sketch-and-precondition appears to be forward stable in practice. No theoretical analysis supporting this finding is known at present.9For those interested, neither iterative sketching nor sketch-and-precondition are backward stable, which is a stronger stability guarantee than forward stability. Fortunately, forward stability is a perfectly adequate stability guarantee for many—but not all—applications.

These conclusions are pretty nuanced. To see what’s going, it can be helpful to look at a graph:10For another randomly generated least-squares problem of the same size with condition number and residual .

The performance of different methods can be summarized as follows: Sketch-and-solve can have very poor forward error. Sketch-and-precondition with the zero initialization is better, but still much worse than the direct method. Iterative sketching and sketch-and-precondition with fair much better, eventually achieving an accuracy comparable to the direct method.

Put more simply, appropriately implemented, sketching works after all!

## Conclusion

Sketching is a computational tool, just like the fast Fourier transform or the randomized SVD. Sketching can be used effectively to solve some problems. But, like any computational tool, sketching is not a silver bullet. Sketching allows you to dimensionality-reduce matrices and vectors, but it distorts them by an appreciable amount. Whether or not this distortion is something you can live with depends on your problem (how much accuracy do you need?) and how you use the sketch (sketch-and-solve or with an iterative method).

# The Hard Way to Prove Jensen’s Inequality

In this post, I want to discuss a very beautiful piece of mathematics I stumbled upon recently. As a warning, this post will be more mathematical than most, but I will still try and sand off the roughest mathematical edges. This post is adapted from a much more comprehensive post by Paata Ivanishvili. My goal is to distill the main idea to its essence, deferring the stochastic calculus until it cannot be avoided.

Jensen’s inequality is one of the most important results in probability.

Jensen’s inequality. Let be a (real) random variable and a convex function such that both and are defined. Then .

Here is the standard proof. A convex function has supporting lines. That is, at a point , there exists a slope such that for all . Invoke this result at and and take expectations to conclude that

In this post, I will outline a proof of Jensen’s inequality which is much longer and more complicated. Why do this? This more difficult proof illustrates an incredible powerful technique for proving inequalities, interpolation. The interpolation method can be used to prove a number of difficult and useful inequalities in probability theory and beyond. As an example, at the end of this post, we will see the Gaussian Jensen inequality, a striking generalization of Jensen’s inequality with many applications.

The idea of interpolation is as follows: Suppose I wish to prove for two numbers and . This may hard to do directly. With the interpolation method, I first construct a family of numbers , , such that and and show that is (weakly) increasing in . This is typically accomplished by showing the derivative is nonnegative:

To prove Jensen’s inequality by interpolation, we shall begin with a special case. As often in probability, the simplest case is that of a Gaussian random variable.

Jensen’s inequality for a Gaussian. Let be a standard Gaussian random variable (i.e., mean-zero and variance ) and let be a thrice-differentiable convex function satisfying a certain technical condition.1Specifically, we assume the regularity condition for some for any Gaussian random variable . Then

Note that the conclusion is exactly Jensen’s inequality, as we have assumed is mean-zero.

The difficulty with any proof by interpolation is to come up with the “right” . For us, the “right” answer will take the form

where starts with no randomness and is our standard Gaussian. To interpolate between these extremes, we increase the variance linearly from to . Thus, we define

Here, and throughout, denotes a Gaussian random variable with zero mean and variance .

Let’s compute the derivative of . To do this, let denote a small parameter which we will later send to zero. For us, the key fact will be that a can be realized as a sum of independent and random variables. Therefore, we write

We now evaluate by using Taylor’s formula

(1)

where lies between and . Now, take expectations,

The random variable has mean zero and variance so this gives

As we show below, the remainder term vanishes as . Thus, we can rearrange this expression to compute the derivative:

The second derivative of a convex function is nonnegative: for every . Therefore,

Jensen’s inequality is proven! In fact, we’ve proven the stronger version of Jensen’s inequality:

This strengthened version can yield improvements. For instance, if is -smooth

then we have

This inequality isn’t too hard to prove directly, but it does show that we’ve obtained something more than the simple proof of Jensen’s inequality.

Analyzing the Remainder Term
Let us quickly check that the remainder term vanishes as . Let’s do this. As an exercise, you can verify that our technical regularity condition implies . Thus, by Hölder’s inequality and setting to be ‘s Hölder conjugate (), we obtain

One can show that where is a function of alone. Therefore, as .

## What’s Really Going On Here?

In our proof, we use a family of random variables , defined for each . Rather than treating these quantities as independent, we can think of them as a collective, comprising a random function known as a Brownian motion.

The Brownian motion is a very natural way of interpolating between a constant and a Gaussian with mean .2The Ornstein–Uhlenbeck process is another natural way of interpolating between a random variable and a Gaussian.

There is an entire subject known as stochastic calculus which allows us to perform computations with Brownian motion and other random processes. The rules of stochastic calculus can seem bizarre at first. For a function of a real number , we often write

For a function of a Brownian motion, the analog is Itô’s formula

While this might seem odd at first, this formula may seem more sensible if we compare with (1) above. The idea, very roughly, is that for an increment of the Brownian motion over a time interval , is a random variable with mean , so we cannot drop the second term in the Taylor series, even up to first order in . Fully diving into the subtleties of stochastic calculus is far beyond the scope of this short post. Hopefully, the rest of this post, which outlines some extensions of our proof of Jensen’s inequality that require more stochastic calculus, will serve as an enticement to learn more about this beautiful subject.

## Proving Jensen by Interpolation

For the rest of this post, we will be less careful with mathematical technicalities. We can use the same idea that we used to prove Jensen’s inequality for a Gaussian random variable to prove Jensen’s inequality for any random variable :

Here is the idea of the proof.

First, realize that we can write any random variable as a function of a standard Gaussian random variable . Indeed, letting and denote the cumulative distribution functions of and , one can show that

has the same distribution as .

Now, as before, we can interpolate between and using a Brownian motion. As a first, idea, we might try

Unfortunately, this choice of does not work! Indeed, does not even equal to ! Instead, we must define

We define using the conditional expectation of the final value conditional on the Brownian motion at an earlier time . Using a bit of elbow grease and stochastic calculus, one can show that

This provides a proof of Jensen’s inequality in general by the method if interpolation.

## Gaussian Jensen Inequality

Now, we’ve come to the real treat, the Gaussian Jensen inequality. In the last section, we saw the sketch of a proof of Jensen’s inequality using interpolation. While it is cool that this proof is possible, we learned anything new since we can prove Jensen’s inequality in other ways. The Gaussian Jensen inequality provides an application of this technique which is hard to prove other ways. This section, in particular, is cribbing quite heavily from Paata Ivanishvili‘s excellent post on the topic.

Here’s the big question:

If are “somewhat dependent”, for which functions does the multivariate Jensen’s inequality

()

hold?

Considering extreme cases, if are entirely dependent, then we would only expect () to hold when is convex. But if are independent, then we can apply Jensen’s inequality to each coordinate one at a time to deduce

We would like a result which interpolates between extremes {fully dependent, fully convex} and {independent, separately convex}. The Gaussian Jensen inequality provides exactly this tool.

As in the previous section, we can generate arbitrary random variables as functions of Gaussian random variables . We will use the covariance matrix of the Gaussian random variables as our measure of the dependence of the random variables . With this preparation in place, we have the following result:

Gaussian Jensen inequality. The conclusion of Jensen’s inequality

(2)

holds for all test functions if and only if

Here, is the Hessian matrix at and denotes the entrywise product of matrices.

This is a beautiful result with striking consequences (see Ivanishvili‘s post). The proof is essentially the same as the proof as Jensen’s inequality by interpolation with a little additional bookkeeping.

Let us confirm this result respects our extreme cases. In the case where are equal (and variance one), is a matrix of all ones and for all . Thus, the Gaussian Jensen inequality states that (2) holds if and only if is positive semidefinite for every , which occurs precisely when is convex.

Next, suppose that are independent and variance one, then is the identity matrix and

A diagonal matrix is positive semidefinite if and only if its entries are nonnegative. Thus, (2) holds if and only if each of ‘s diagonal second derivatives are nonnegative : this is precisely the condition for to be separately convex in each argument.

There’s much more to be said about the Gaussian Jensen inequality, and I encourage you to read Ivanishvili‘s post to see the proof and applications. What I find so compelling about this result—so compelling that I felt the need to write this post—is how interpolation and stochastic calculus can be used to prove inequalities which don’t feel like stochastic calculus problems. The Gaussian Jensen inequality is a statement about functions of dependent Gaussian random variables; there’s nothing dynamic happening. Yet, to prove this result, we inject dynamics into the problem, viewing the two sides of our inequality as endpoints of a random process connecting them. This is a such a beautiful idea that I couldn’t help but share it.

# Stochastic Trace Estimation

I am delighted to share that me, Joel A. Tropp, and Robert J. Webber‘s paper XTrace: Making the Most of Every Sample in Stochastic Trace Estimation has recently been released as a preprint on arXiv. In it, we consider the implicit trace estimation problem:

Implicit trace estimation problem: Given access to a square matrix via the matrix–vector product operation , estimate its trace .

Algorithms for this task have many uses such as log-determinant computations in machine learning, partition function calculations in statistical physics, and generalized cross validation for smoothing splines. I described another application to counting triangles in a large network in a previous blog post.

Our paper presents new trace estimators XTrace and XNysTrace which are highly efficient, producing accurate trace approximations using a small budget of matrix–vector products. In addition, these algorithms are fast to run and are supported by theoretical results which explain their excellent performance. I really hope that you will check out the paper to learn more about these estimators!

For the rest of this post, I’m going to talk about the most basic stochastic trace estimation algorithm, the GirardHutchinson estimator. This seemingly simple algorithm exhibits a number of nuances and forms the backbone for more sophisticated trace estimates such as Hutch++, Nyström++, XTrace, and XNysTrace. Toward the end, this blog post will be fairly mathematical, but I hope that the beginning will be fairly accessible to all.

## Girard–Hutchinson Estimator: The Basics

The GirardHutchinson estimator for the trace of a square matrix is

(1)

Here, are random vectors, usually chosen to be statistically independent, and denotes the conjugate transpose of a vector or matrix. The Girard–Hutchinson estimator only depends on the matrix through the matrix–vector products .

### Unbiasedness

Provided the random vectors are isotropic

(2)

the Girard–Hutchinson estimator is unbiased:

(3)

Let us confirm this claim in some detail. First, we use linearity of expectation to evaluate

(4)

Therefore, to prove that , it is sufficient to prove that for each .

When working with traces, there are two tricks that solve 90% of derivations. The first trick is that, if we view a number as a matrix, then a number equals its trace, . The second trick is the cyclic property: For a matrix and a matrix , we have . The cyclic property should be handled with care when one works with a product of three or more matrices. For instance, we have

However,

One should think of the matrix product as beads on a closed loop of string. One can move the last bead to the front of the other two, , but not interchange two beads, .

With this trick in hand, let’s return to proving that for every . Apply our two tricks:

The expectation is a linear operation and the matrix is non-random, so we can bring the expectation into the trace as

Invoke the isotropy condition (2) and conclude:

Plugging this into (4) confirms the unbiasedness claim (3).

### Variance

Continue to assume that the ‘s are isotropic (3) and now assume that are independent. By independence, the variance can be written as

Assuming that are identically distributed , we then get

The variance decreases like , which is characteristic of Monte Carlo-type algorithms. Since is unbiased (i.e, (3)), this means that the mean square error decays like so the average error (more precisely root-mean-square error) decays like

This type of convergence is very slow. If I want to decrease the error by a factor of , I must do the work!

Variance-reduced trace estimators like Hutch++ and our new trace estimator XTrace improve the rate of convergence substantially. Even in the worst case, Hutch++ and XTrace reduce the variance at a rate and (root-mean-square) error at rates :

For matrices with rapidly decreasing singular values, the variance and error can decrease much faster than this.

## Variance Formulas

As the rate of convergence for the Girard–Hutchinson estimator is so slow, it is imperative to pick a distribution on test vectors that makes the variance of the single–sample estimate as low as possible. In this section, we will provide several explicit formulas for the variance of the Girard–Hutchinson estimator. Derivations of these formulas will appear at the end of this post. These variance formulas help illuminate the benefits and drawbacks of different test vector distributions.

To express the formulas, we will need some notation. For a complex number we use and to denote the real and imaginary parts. The variance of a random complex number is

The Frobenius norm of a matrix is

If is real symmetric or complex Hermitian with (real) eigenvalues , we have

(5)

denotes the ordinary transpose of and denotes the conjugate transpose of .

### Real-Valued Test Vectors

We first focus on real-valued test vectors . Since is real, we can use the ordinary transpose rather than the conjugate transpose . Since is a number, it is equal to its own transpose:

Therefore,

The Girard–Hutchinson trace estimator applied to is the same as the Girard–Hutchinson estimator applied to the symmetric part of , .

For the following results, assume is symmetric, .

1. Real Gaussian: are independent standard normal random vectors.

2. Uniform signs (Rademachers): are independent random vectors with uniform coordinates.

3. Real sphere: Assume are uniformly distributed on the real sphere of radius : .

These formulas continue to hold for nonsymmetric by replacing by its symmetric part on the right-hand sides of these variance formulas.

### Complex-Valued Test Vectors

We now move our focus to complex-valued test vectors . As a rule of thumb, one should typically expect that the variance for complex-valued test vectors applied to a real symmetric matrix is about half the natural real counterpart—e.g., for complex Gaussians, you get about half the variance than with real Gaussians.

A square complex matrix has a Cartesian decomposition

where

denote the Hermitian and skew-Hermitian parts of . Similar to how the imaginary part of a complex number is real, the skew-Hermitian part of a complex matrix is Hermitian (and is skew-Hermitian). Since and are both Hermitian, we have

Consequently, the variance of can be broken into Hermitian and skew-Hermitian parts:

For this reason, we will state the variance formulas only for Hermitian , with the formula for general following from the Cartesian decomposition.

For the following results, assume is Hermitian, .

1. Complex Gaussian: are independent standard complex random vectors, i.e., each has iid entries distributed as for standard normal random variables.

2. Uniform phases (Steinhauses): are independent random vectors whose entries are uniform on the complex unit circle .

3. Complex sphere: Assume are uniformly distributed on the complex sphere of radius : .

## Optimality Properties

Let us finally address the question of what the best choice of test vectors is for the Girard–Hutchinson estimator. We will state two results with different restrictions on .

Our first result, due to Hutchinson, is valid for real symmetric matrices with real test vectors.

Optimality (independent test vectors with independent coordinates). If the test vectors are isotropic (2), independent from each other, and have independent entries, then for any fixed real symmetric matrix , the minimum variance for is obtained when are populated with random signs .

The next optimality results will have real and complex versions. To present the results for -valued and an -valued test vectors on unified footing, let denote either or . We let a -Hermitian matrix be either a real symmetric matrix (if ) or a complex Hermitian matrix (if ). Let a -unitary matrix be either a real orthogonal matrix (if ) or a complex unitary matrix (if ).

The condition that the vectors have independent entries is often too restrictive in practice. It rules out, for instance, the case of uniform vectors on the sphere. If we relax this condition, we get a different optimal distribution:

Optimality (independent test vectors). Consider any set of -Hermitian matrices which is invariant under -unitary similary transformations:

Assume that the test vectors are independent and isotropic (2). The worst-case variance is minimized by choosing uniformly on the -sphere: .

More simply, if you wants your stochastic trace estimator to be effective for a class of inputs (closed under -unitary similarity transformations) rather than a single input matrix , then the best distribution are test vectors drawn uniformly from the sphere. Examples of classes of matrices include:

• Fixed eigenvalues. For fixed real eigenvalues , the set of all -Hermitian matrices with these eigenvalues.
• Density matrices. The class of all trace-one psd matrices.
• Frobenius norm ball. The class of all -Hermitian matrices of Frobenius norm at most 1.

## Derivation of Formulas

In this section, we provide derivations of the variance formulas. I have chosen to focus on derivations which are shorter but use more advanced techniques rather than derivations which are longer but use fewer tricks.

### Real Gaussians

First assume is real. Since is real symmetric, has an eigenvalue decomposition , where is orthogonal and is a diagonal matrix reporting ‘s eigenvalues. Since the real Gaussian distribution is invariant under orthogonal transformations, has the same distribution as . Therefore,

Here, we used that the variance of a squared standard normal random variable is two.

For non-real matrix, we can break the matrix into its entrywise real and imaginary parts . Thus,

### Uniform Signs

First, compute

For a vector of uniform random signs, we have for every , so the second sum vanishes. Note that we have assumed symmetric, so the sum over can be replaced by two times the sum over :

Note that are pairwise independent. As a simple exercise, one can verify that the identity

holds for any pairwise independent family of random variances and numbers . Ergo,

In the second-to-last line, we use the fact that is a uniform random sign, which has variance . The final line is a consequence of the symmetry of .

### Uniform on the Real Sphere

The simplest proof is I know is by the “camel principle”. Here’s the story (a lightly edited quotation from MathOverflow):

A father left 17 camels to his three sons and, according to the will, the eldest son was to be given a half of the camels, the middle son one-third, and the youngest son the one-ninth. The sons did not know what to do since 17 is not evenly divisible into either two, three, or nine parts, but a wise man helped the sons: he added his own camel, the oldest son took camels, the second son took camels, the third son camels and the wise man took his own camel and went away.

We are interested in a vector which is uniform on the sphere of radius . Performing averages on the sphere is hard, so we add a camel to the problem by “upgrading” to a spherically symmetric vector which has a random length. We want to pick a distribution for which the computation is easy. Fortunately, we already know such a distribution, the Gaussian distribution, for which we already calculated .

The Gaussian vector and the uniform vector on the sphere are related by

where is the squared length of the Gaussian vector . In particular, has the distribution of the sum of squared Gaussian random variables, which is known as a random variable with degrees of freedom.

Now, we take the camel back. Compute the variance of using the chain rule for variance:

Here, and denote the conditional variance and conditional expectation with respect to the random variable . The quick and dirty ways of working with these are to treat the random variable “like a constant” with respect to the conditional variance and expectation.

Plugging in the formula and treating “like a constant”, we obtain

As we mentioned, is a random variable with degrees of freedom and and are known quantities that can be looked up:

We know and . Plugging these all in, we get

Rearranging, we obtain

### Complex Gaussians

The trick is the same as for real Gaussians. By invariance of complex Gaussian random vectors under unitary transformations, we can reduce to the case where is a diagonal matrix populated with eigenvalues . Then

Here, we use the fact that is a random variable with two degrees of freedom, which has variance four.

### Random Phases

The trick is the same as for uniform signs. A short calculation (remembering that is Hermitian and thus ) reveals that

The random variables are pairwise independent so we have

Since is uniformly distributed on the complex unit circle, we can assume without loss of generality that . Thus, letting be uniform on the complex unit circle,

The real and imaginary parts of have the same distribution so

so . Thus

### Uniform on the Complex Sphere: Derivation 1 by Reduction to Real Case

There are at least three simple ways of deriving this result: the camel trick, reduction to the real case, and Haar integration. Each of these techniques illustrates a trick that is useful in its own right beyond the context of trace estimation. Since we have already seen an example of the camel trick for the real sphere, I will present the other two derivations.

Let us begin with the reduction to the real case. Let and denote the real and imaginary parts of a vector or matrix, taken entrywise. The key insight is that if is a uniform random vector on the complex sphere of radius , then

We’ve converted the complex vector into a real vector .

Now, we need to convert the complex matrix into a real matrix . To do this, recall that one way of representing complex numbers is by matrices:

Using this correspondence addition and multiplication of complex numbers can be carried by addition and multiplication of the corresponding matrices.

To convert complex matrices to real matrices, we use a matrix-version of the same representation:

One can check that addition and multiplication of complex matrices can be carried out by addition and multiplication of the corresponding “realified” matrices, i.e.,

holds for all complex matrices and .

We’ve now converted complex matrix and vector into real matrix and vector . Let’s compare to . A short calculation reveals

Since is a uniform random vector on the sphere of radius , is a uniform random vector on the sphere of radius . Thus, by the variance formula for the real sphere, we get

A short calculation verifies that and . Plugging this in, we obtain

### Uniform on the Complex Sphere: Derivation 2 by Haar Integration

The proof by reduction to the real case requires some cumbersome calculations and requires that we have already computed the variance in the real case by some other means. The method of Haar integration is more slick, but it requires some pretty high-power machinery. Haar integration may be a little bit overkill for this problem, but this technique is worth learning as it can handle some truly nasty expected value computations that appear, for example, in quantum information.

We seek to compute

The first trick will be to write this expession using a single matrix trace using the tensor (Kronecker) product . For those unfamiliar with the tensor product, the main properties we will be using are

(6)

We saw in the proof of unbiasedness that

Therefore, by (6),

Thus, to evaluate , it will be sufficient to evaluate . Forunately, there is a useful formula for these expectation provided by a field of mathematics known as representation theory (see Lemma 1 in this paper):

Here, is the orthogonal projection onto the space of symmetric two-tensors . Therefore, we have that

To evalute the trace on the right-hand side of this equation, there is another formula (see Lemma 6 in this paper):

Therefore, we conclude

## Proof of Optimality Properties

In this section, we provide proofs of the two optimality properties.

### Optimality: Independent Vectors with Independent Coordinates

Assume is real and symmetric and suppose that is isotropic (2) with independent coordinates. The isotropy condition

implies that , where is the Kronecker symbol. Using this fact, we compute the second moment:

Thus

The variance is minimized by choosing with as small as possible. Since , the smallest possible value for is , which is obtained by populating with random signs.

### Optimality: Independent Vectors

This result appears to have first been proven by Richard Kueng in unpublished work. We use an argument suggested to me by Robert J. Webber.

Assume is a class of -Hermitian matrices closed under -unitary similarity transformations and that is an isotropic random vector (2). Decompose the test vector as

First, we shall show that the variance is reduced by replacing with a vector drawn uniformly from the sphere

(7)

where

(8)

Note that such a can be generated as for a uniformly random -unitary matrix . Therefore, we have

Now apply Jensen’s inequality only over the randomness in to obtain

Finally, note that since is closed under -unitary similarity transformations, the supremum over for is the same as the supremum of , so we obtain

We have successfully proven (7). This argument is a specialized version of a far more general result which appears as Proposition 4.1 in this paper.

Next, we shall prove

(9)

where is still defined as in (8). Indeed, using the chain rule for variance, we obtain

Here, we have used that is uniform on the sphere and thus . By definition, is the length of divided by . Therefore,

Therefore, by Jensen’s inequality,

Thus

which proves (9).

# Chebyshev Polynomials

This post is co-written by my brother, Aidan Epperly, for the second Summer of Math Exposition (SoME2).

Let’s start with a classical problem: connect-the-dots. As we know from geometry, any two points in the plane are connected by one and only one straight line:

But what if we have more than two points? How should we connect them? One natural way is by parabola. Any three points (with distinct coordinates) are connected by one and only one parabola :

And we can keep extending this. Any points1The degree of the polynomial is one less than the number of points because a degree- polynomial is described by coefficients. For instance, a degree-two parabola has three coefficients , , and . (with distinct coordinates) are connected by a unique degree- polynomial :

This game of connect-the-dots with polynomials is known more formally as polynomial interpolation. We can use polynomial interpolation to approximate functions. For instance, we can approximate the function on the interval to visually near-perfect accuracy by connecting the dots between seven points :

But something very peculiar happens when we try and apply this trick to the specially chosen function on the interval :

Unlike , the polynomial interpolant for is terrible! What’s going on? Why doesn’t polynomial interpolation work here? Can we fix it? The answer to the last question is yes and the solution is Chebyshev polynomials.

## Reverse-Engineering Chebyshev

The failure of polynomial interpolation for is known as Runge’s phenomenon after Carl Runge who discovered this curious behavior in 1901. The function is called the Runge function. Our goal is to find a fix for polynomial interpolation which crushes the Runge phenomenon, allowing us to reliably approximate every sensible2A famous theorem of Faber states that there does not exist any set of points through which the polynomial interpolants converge for every continuous function. This is not as much of a problem as it may seem. As the famous Weierstrass function shows, arbitrary continuous functions can be very weird. If we restrict ourselves to nicer functions, such as Lipschitz continuous functions, there does exist a set of points through which the polynomial interpolant always converges to the underlying function. Thus, in this senses, it is possible to crush the Runge phenomenon. function with polynomial interpolation.

Carl Runge

Let’s put on our thinking caps and see if we can discover the fix for ourselves. In order to discover a fix, we must first identify the problem. Observe that the polynomial interpolant is fine near the center of the interval; it only fails near the boundary.

This leads us to a guess for what the problem might be; maybe we need more interpolation points near the boundaries of the interval. Indeed, tipping our hand a little bit, this turns out to be the case. For instance, connecting the dots for the following set of “mystery points” clustered at the endpoints works just fine:

Let’s experiment a little and see if we can discover a nice set of interpolation points, which we will call , like this for ourselves. We’ll assume the interpolation points are given by a function so we can form the polynomial interpolant for any desired polynomial degree .3Technically, we should insist on the function being \textit{injective} so that the points are guaranteed to be distinct. For instance, if we pick , the points look like this:

Equally spaced points (shown on vertical axis) give rise to
non-equally spaced points (shown on horizontal axis)

How should we pick the function ? First observe that, even for the Runge function, equally spaced interpolation points are fine near the center of the interval. We thus have at least two conditions for our desired interpolation points:

1. The interior points should maintain their spacing of roughly .
2. The points must cluster near both boundaries.

As a first attempt let’s divide the interval into thirds and halve the spacing of points except in the middle third. This leads to the function

These interpolation points initially seem promising, even successfully approximating the Runge function itself.

Unfortunately, this set of points fails when we consider other functions. For instance, if we use the Runge-like function , we see that these interpolation points now lead to a failure to approximate the function at the middle of the interval, even if we use a lot of interpolation points!

Maybe the reason this set of interpolation points didn’t work is that the points are too close at the endpoints. Or maybe we should have divided the interval as quarter–half–quarter rather than thirds. There are lots of variations of this strategy for choosing points to explore and all of them eventually lead to failure on some Runge-flavored example. We need a fundamentally different strategy then making the points times closer within distance of the endpoints.

Let’s try a different approach. The closeness of the points at the endpoints is determined by the slope of the function at and . The smaller that and are, the more clustered the points will be. For instance,

So if we want the points to be much more clustered together, it is natural to require

It also makes sense for the function to cluster points equally near both endpoints, since we see no reason to preference one end over the other. Collecting together all the properties we want the function to have, we get the following list:

1. spans the whole range ,
2. , and
3. is symmetric about , .

Mentally scrolling through our Rolodex of friendly functions, a natural one that might come to mind meeting these three criteria is the cosine function, specifically . This function yields points which are more clustered at the endpoints:

The points

we guessed our way into are known as the Chebyshev points.4Some authors refer to these as the “Chebyshev points of the second kind” or use other names. We follow the convention in Approximation Theory and Approximation Practice (Chapter 1) and simply refer to these points simply as the Chebyshev points. The Chebyshev points prove themselves perfectly fine for the Runge function:

As we saw earlier, success on the Runge function alone is not enough to declare victory for the polynomial interpolation problem. However, in this case, there are no other bad examples left to find. For any nice function with no jumps, polynomial interpolation through the Chebyshev points works excellently.5Specifically, for a function which not too rough (i.e., Lipschitz continuous), the degree- polynomial interpolant of through the Chevyshev points converges uniformly to as .

## Why the Chebyshev Points?

We’ve guessed our way into a solution to the polynomial interpolation problem, but we still really don’t know what’s going on here. Why are the Chebyshev points much better at polynomial interpolation than equally spaced ones?

Now that we know that the Chebyshev points are a right answer to the interpolation problem,6Indeed, there are other sets of interpolation points through which polynomial interpolation also works well, such as the Legendre points. let’s try and reverse engineer a principled reason for why we would expect them to be effective for this problem. To do this, we ask:

What is special about the cosine function?

From high school trigonometry, we know that gives the coordinate of a point radians along the unit circle. This means that the Chebyshev points are the coordinates of equally spaced points on the unit circle (specifically the top half of the unit circle ).

Chebyshev points are the coordinates of equally spaced points on the unit circle.

This raises the question:

What does the interpolating polynomial look like as a function of the angle ?

To convert between and we simply plug in to :

This new function depending on , which we can call , is a polynomial in the variable . Powers of cosines are not something we encounter every day, so it makes sense to try and simplify things using some trig identities. Here are the first couple powers of cosines:

A pattern has appeared! The powers always take the form7As a fun exercise, you might want to try and prove this using mathematical induction.

The significance of this finding is that, by plugging in each of these formulas for , we see that our polynomial in the variable has morphed into a Fourier cosine series in the variable :

For anyone unfamiliar with Fourier series, we highly encourage the 3Blue1Brown video on the subject, which explains why Fourier series are both mathematically beautiful and practically useful. The basic idea is that almost any function can be expressed as a combination of waves (that is, sines and cosines) with different frequencies.8More precisely, we might call these angular frequencies. In our case, this formula tells us that is equal to units of frequency , plus units of frequency , all the way up to units of frequency . Different types of Fourier series are appropriate in different contexts. Since our Fourier series only possesses cosines, we call it a Fourier cosine series.

We’ve discovered something incredibly cool:

Polynomial interpolation through the Chebyshev points is equivalent to finding a Fourier cosine series for equally spaced angles .

We’ve arrived at an answer to why the Chebyshev points work well for polynomial interpolation.

Polynomial interpolation through the Chebyshev points is effective because Fourier cosine series through equally spaced angles is effective.

Of course, this explanation just raises the further question: Why do Fourier cosine series give effective interpolants through equally spaced angles ? This question has a natural answer as well, involving the convergence theory and aliasing formula (see Section 3 of this paper) for Fourier series. We’ll leave the details to the interested reader for investigation. The success of Fourier cosines series in interpolating equally spaced data is a fundamental observation that underlies the field of digital signal processing. Interpolation through the Chebyshev points effectively hijacks this useful fact and applies it to the seemingly unrelated problem of polynomial interpolation.

Another question this explanation raises is the precise meaning of “effective”. Just how good are polynomial interpolants through the Chebyshev points at approximating functions? As is discussed at length in another post on this blog, the degree to which a function can be effectively approximated is tied to how smooth or rough it is. Chebyshev interpolants approximate nice analytic functions like or with exponentially small errors in the number of interpolation points used. By contrast, functions with kinks like are approximated with errors which decay much more slowly. See theorems 2 and 3 on this webpage for more details.

## Chebyshev Polynomials

We’ve now discovered a set of points, the Chebyshev points, through which polynomial interpolation works well. But how should we actually compute the interpolating polynomial

Again, it will be helpful to draw on the connection to Fourier series. Computations with Fourier series are highly accurate and can be made lightning fast using the fast Fourier transform algorithm. By comparison, directly computing with a polynomial through its coefficients is a computational nightmare.

In the variable , the interpolant takes the form

To convert back to , we use the inverse function9One always has to be careful when going from to since multiple values get mapped to a single value by the cosine function. Fortunately, we’re working with variables and , between which the cosine function is one-to-one with the inverse function being given by the arccosine. to obtain:

This is a striking formula. Given all of the trigonometric functions, it’s not even obvious that is a polynomial (it is)!

Despite its seeming peculiarity, this is a very powerful way of representing the polynomial . Rather than expressing using monomials , we’ve instead written as a combination of more exotic polynomials

The polynomials are known as the Chebyshev polynomials,10More precisely, the polynomials are known as the Chebyshev polynomials of the first kind. named after Pafnuty Chebyshev who studied the polynomials intensely.11The letter “T” is used for Chebyshev polynomials since the Russian name “Chebyshev” is often alternately transliterated to English as “Tchebychev”.

Pafnuty Chebyshev

Writing out the first few Chebyshev polynomials shows they are indeed polynomials:

The first four Chebyshev polynomials

To confirm that this pattern does continue, we can use trig identities to derive12Specifically, the recurrence is a consequence of applying the sum-to-product identity to for . the following recurrence relation for the Chebyshev polynomials:

Since and are both polynomials, every Chebyshev polynomial is as well.

We’ve arrived at the following amazing conclusion:

Under the change of variables , the Fourier cosine series

becomes the combination of Chebyshev polynomials

This simple and powerful observations allows us to apply the incredible speed and accuracy of Fourier series to polynomial interpolation.

Beyond being a neat idea with some nice mathematics, this connection between Fourier series and Chebyshev polynomials is a powerful tool for solving computational problems. Once we’ve accurately approximated a function by a polynomial interpolant, many quantities of interest (derivatives, integrals, zeros) become easy to compute—after all, we just have to compute them for a polynomial! We can also use Chebyshev polynomials to solve differential equations with much faster rates of convergence than other methods. Because of the connection to Fourier series, all of these computations can be done to high accuracy and blazingly fast via the fast Fourier transform, as is done in the software package Chebfun.

The Chebyshev polynomials have an array of amazing properties and they appear all over mathematics and its applications in other fields. Indeed, we have only scratched the surface of the surface. Many questions remain:

• What is the connection between the Chebyshev points and the Chebyshev polynomials?
• The cosine functions are orthogonal to each other; are the Chebyshev polynomials?
• Are the Chebyshev points the best points for polynomial interpolation? What does “best” even mean in this context?
• Every “nice” even periodic function has an infinite Fourier cosine series which converges to it. Is there a Chebyshev analog? Is there a relation between the infinite Chebyshev series and the (finite) interpolating polynomial through the Chebyshev points?

All of these questions have beautiful and fairly simple answers. The book Approximation Theory and Approximation Practice is a wonderfully written book that answers all of these questions in its first six chapters, which are freely available on the author’s website. We recommend the book highly to the curious reader.

TL;DR: To get an accurate polynomial approximation, interpolate through the Chebyshev points.
To compute the resulting polynomial, change variables to , compute the Fourier cosine series interpolant, and obtain your polynomial interpolant as a combination of Chebyshev polynomials.