Low-Rank Approximation Toolbox: Nyström, Cholesky, and Schur

October 24, 2022 by Ethan Epperly Leave a comment

In the last post, we looked at the Nyström approximation to an $N\times N$ positive semidefinite (psd) matrix $A$ . A special case was the column Nyström approximation, defined to be¹

(Nys) $\hat{A} = A(:,S) \, A(S,S)^{-1} \, A(S,:),$

where $S = \{s_1,\ldots,s_k\} \subseteq \{1,2,\ldots,N\}$ identifies a subset of $k$ columns of $A$ . Provided $k\ll N$ , this allowed us to approximate all $N^2$ entries of the matrix $A$ using only the $kN$ entries in columns $s_1,\ldots,s_k$ of $A$ , a huge savings of computational effort!

With the column Nyström approximation presented just as such, many questions remain:

Why this formula?
Where did it come from?
How do we pick the columns $s_1,\ldots,s_k$ ?
What is the residual $A - \hat{A}$ of the approximation?

In this post, we will answer all of these questions by drawing a connection between low-rank approximation by Nyström approximation and solving linear systems of equations by Gaussian elimination. The connection between these two seemingly unrelated areas of matrix computations will pay dividends, leading to effective algorithms to compute Nyström approximations by the (partial) Cholesky factorization of a positive (semi)definite matrix and an elegant description of the residual of the Nyström approximation as the Schur complement.

Cholesky: Solving Linear Systems

Suppose we want solve the system of linear equations $Ax = b$ , where $A$ is a real $N\times N$ invertible matrix and $b$ is a vector of length $N$ . The standard way of doing this in modern practice (at least for non-huge matrices $A$ ) is by means of Gaussian elimination/LU factorization. We factor the matrix $A$ as a product $A = LU$ of a lower triangular matrix $L$ and an upper triangular matrix $U$ .² The system $Ax = b$ is solved by first solving $Ly = b$ for $y$ and then $Ux = y$ for $x$ ; the triangularity of $L$ and $U$ make solving the associated systems of linear equations easy.

For real symmetric positive definite matrix $A$ , a simplification is possible. In this case, one can compute an LU factorization where the matrices $L$ and $U$ are transposes of each other, $U = L^\top$ . This factorization $A = LL^\top$ is known as a Cholesky factorization of the matrix $A$ .

The Cholesky factorization can be easily generalized to allow the matrix $A$ to be complex-valued. For a complex-valued positive definite matrix $A$ , its Cholesky decomposition takes the form $A = LL^*$ , where $L$ is again a lower triangular matrix. All that has changed is that the transpose ${}^\top$ has been replaced by the conjugate transpose ${}^*$ . We shall work with the more general complex case going forward, though the reader is free to imagine all matrices as real and interpret the operation ${}^*$ as ordinary transposition if they so choose.

Schur: Computing the Cholesky Factorization

Here’s one way of computing the Cholesky factorization using recursion. Write the matrix $A$ in block form as

$A = \twobytwo{A_{11}}{A_{12}}{A_{21}}{A_{22}}.$

Our first step will be “block Cholesky factorize” the matrix $A$ , factoring $A$ as a product of matrices which are only block triangular. Then, we’ll “upgrade” this block factorization into a full Cholesky factorization.

The core idea of Gaussian elimination is to combine rows of a matrix to introduce zero entries. For our case, observe that multiplying the first block row of $A$ by $A_{21}A_{11}^{-1}$ and subtracting this from the second block row introduces a matrix of zeros into the bottom left block of $A$ . (The matrix $A_{11}$ is a principal submatrix of $A$ and is therefore guaranteed to be positive definite and thus invertible.³) In matrix language,

$\twobytwo{I}{0}{-A_{21}A_{11}^{-1}}{I}\twobytwo{A_{11}}{A_{12}}{A_{21}}{A_{22}} = \twobytwo{A_{11}}{A_{12}}{0}{A_{22} - A_{21}A_{11}^{-1}A_{12}}.$

Isolating $A$ on the left-hand side of this equation by multiplying by

$\twobytwo{I}{0}{-A_{21}A_{11}^{-1}}{I}^{-1} = \twobytwo{I}{0}{A_{21}A_{11}^{-1}}{I}$

yields the block triangular factorization

$A = \twobytwo{A_{11}}{A_{12}}{A_{21}}{A_{22}} = \twobytwo{I}{0}{A_{21}A_{11}^{-1}}{I} \twobytwo{A_{11}}{A_{12}}{0}{A_{22} - A_{21}A_{11}^{-1}A_{12}}.$

We’ve factored $A$ into block triangular pieces, but these pieces are not (conjugate) transposes of each other. Thus, to make this equation more symmetrical, we can further decompose

(1) $A = \twobytwo{A_{11}}{A_{12}}{A_{21}}{A_{22}} = \twobytwo{I}{0}{A_{21}A_{11}^{-1}}{I} \twobytwo{A_{11}}{0}{0}{A_{22} - A_{21}A_{11}^{-1}A_{12}} \twobytwo{I}{0}{A_{21}A_{11}^{-1}}{I}^*.$

This is a block version of the Cholesky decomposition of the matrix $A$ taking the form $A = LDL^*$ , where $L$ is a block lower triangular matrix and $D$ is a block diagonal matrix.

We’ve met the second of our main characters, the Schur complement

(Sch) $S = A_{22} - A_{21}A_{11}^{-1}A_{12}.$

This seemingly innocuous combination of matrices has a tendency to show up in surprising places when one works with matrices.⁴ It’s appearance in any one place is unremarkable, but the shear ubiquity of $A_{22} - A_{21}A_{11}^{-1}A_{12}$ in matrix theory makes it deserving of its special name, the Schur complement. To us for now, the Schur complement is just the matrix appearing in the bottom right corner of our block Cholesky factorization.

The Schur complement enjoys the following property:⁵

Positivity of the Schur complement: If $A=\twobytwo{A_{11}}{A_{12}}{A_{21}}{A_{22}}$ is positive (semi)definite, then the Schur complement $S = A_{22} - A_{21}A_{11}^{-1}A_{12}$ is positive (semi)definite.

As a consequence of this property, we conclude that both $A_{11}$ and $S$ are positive definite.

With the positive definiteness of the Schur complement in hand, we now return to our Cholesky factorization algorithm. Continue by recursively⁶ computing Cholesky factorizations of the diagonal blocks

$A_{11} = L_{11}^{\vphantom{*}}L_{11}^*, \quad S = L_{22}^{\vphantom{*}}L_{22}^*.$

Inserting these into the block $LDL^*$ factorization (1) and simplifying gives a Cholesky factorization, as desired:

$A = \twobytwo{L_{11}}{0}{A_{21}^{\vphantom{*}}(L_{11}^{*})^{-1}}{L_{22}}\twobytwo{L_{11}}{0}{A_{21}^{\vphantom{*}}(L_{11}^{*})^{-1}}{L_{22}}^* =: LL^*.$

Voilà, we have obtained a Cholesky factorization of a positive definite matrix $A$ !

By unwinding the recursions (and always choosing the top left block $A_{11}$ to be of size $1\times 1$ ), our recursive Cholesky algorithm becomes the following iterative algorithm: Initialize $L$ to be the zero matrix. For $j = 1,2,3,\ldots,N$ , perform the following steps:

Update $L$ . Set the $j$ th column of $L$ :
$L(j:N,j) \leftarrow A(j:N,j)/\sqrt{a_{jj}}.$
Update $A$ . Update the bottom right portion of $A$ to be the Schur complement:
$A(j+1:N,j+1:N)\leftarrow A(j+1:N,j+1:N) - \frac{A(j+1:N,j)A(j,j+1:N)}{a_{jj}}.$

This iterative algorithm is how Cholesky factorization is typically presented in textbooks.

Nyström: Using Cholesky Factorization for Low-Rank Approximation

Our motivating interest in studying the Cholesky factorization was the solution of linear systems of equations $Ax = b$ for a positive definite matrix $A$ . We can also use the Cholesky factorization for a very different task, low-rank approximation.

Let’s first look at things through the lense of the recursive form of the Cholesky factorization. The first step of the factorization was to form the block Cholesky factorization

$A = \twobytwo{I}{0}{A_{21}A_{11}^{-1}}{I} \twobytwo{A_{11}}{0}{0}{A_{22} - A_{21}A_{11}^{-1}A_{12}} \twobytwo{I}{0}{A_{21}A_{11}^{-1}}{I}^*.$

Suppose that we choose the top left $A_{11}$ block to be of size $k\times k$ , where $k$ is much smaller than $N$ . The most expensive part of the Cholesky factorization will be the recursive factorization of the Schur complement $A_{22} - A_{21}A_{11}^{-1} A_{12}$ , which is a large matrix of size $(N-k)\times (N-k)$ .

To reduce computational cost, we ask the provocative question: What if we simply didn’t factorize the Schur complement? Observe that we can write the block Cholesky factorization as a sum of two terms

(2) $A = \twobyone{I}{A_{21}{A_{11}^{-1}}} A_{11} \twobyone{I}{A_{22}{A_{11}^{-1}}}^* + \twobytwo{0}{0}{0}{A_{22}-A_{21}A_{11}^{-1}A_{12}}.$

We can use the first term of this sum as a rank- $k$ approximation to the matrix $A$ . The low-rank approximation, which we can write out more conveniently as

$\hat{A} = \twobyone{A_{11}}{A_{21}} A_{11}^{-1} \onebytwo{A_{11}}{A_{12}} = \twobytwo{A_{11}}{A_{12}}{A_{21}}{A_{21}A_{11}^{-1}A_{12}},$

is the column Nyström approximation (Nys) to $A$ associated with the index set $S = \{1,2,3,\ldots,k\}$ and is the final of our three titular characters. The residual of the Nyström approximation is the second term in (2), which is none other than the Schur complement (Sch), padded by rows and columns of zeros:

$A - \hat{A} = \twobytwo{0}{0}{0}{A_{22}-A_{21}A_{11}^{-1}A_{12}}.$

Observe that the approximation $\hat{A}$ is obtained from the process of terminating a Cholesky factorization midway through algorithm execution, so we say that the Nyström approximation results from a partial Cholesky factorization of the matrix $A$ .

Summing things up:

If we perform a partial Cholesky factorization on a positive (semi)definite matrix, we obtain a low-rank approximation known as the column Nyström approximation. The residual of this approximation is the Schur complement, padded by rows and columns of zeros.

The idea of obtaining a low-rank approximation from a partial matrix factorization is very common in matrix computations. Indeed, the optimal low-rank approximation to a real symmetric matrix is obtained by truncating its eigenvalue decomposition and the optimal low-rank approximation to a general matrix is obtained by truncating its singular value decomposition. While the column Nyström approximation is not the optimal rank- $k$ approximation to $A$ (though it does satisfy a weaker notion of optimality, as discussed in this previous post), it has a killer feature not possessed by the optimal approximation:

The column Nyström approximation is formed from only $k$ columns from the matrix $A$ . A column Nyström approximation approximates an $N\times N$ matrix by only reading a fraction of its entries!

Unfortunately there’s not a free lunch here. The column Nyström is only a good low-rank approximation if the Schur complement has small entries. In general, this need not be the case. Fortunately, we can improve our situation by means of pivoting.

Our iterative Cholesky algorithm first performs elimination using the entry in position $(1,1)$ followed by position $(2,2)$ and so on. There’s no need to insist on this exact ordering of elimination steps. Indeed, at each step of the Cholesky algorithm, we can choose whichever diagonal position $(j,j)$ that we want to perform elimination. The entry we choose to perform elimination with is called a pivot.

Obtaining good column Nyström approximations requires identifying the a good choice for the pivots to reduce the size of the entries of the Schur complement at each step of the algorithm. With general pivot selection, an iterative algorithm for computing a column Nyström approximation by partial Cholesky factorization proceeds as follows: Initialize an $N\times k$ matrix $F$ to store the column Nyström approximation $\hat{A} = FF^*$ , in factored form. For $j = 1,2,\ldots,k$ , perform the following steps:

Select pivot. Choose a pivot $s_j$ .
Update the approximation. $F(:,j) \leftarrow A(:,s_j) / \sqrt{a_{s_js_j}}$ .
Update the residual. $A \leftarrow A - \frac{A(:,s_j)A(s_j,:)}{a_{s_js_j}}$ .

This procedure results in the Nyström approximation (Nys) with column set $S = \{s_1,\ldots,s_k\}$ :

$\hat{A} = FF^* = A(:,S) \, A(S,S)^{-1} \, A(S,:).$

The pivoted Cholesky steps 1–3 requires updating the entire matrix $A$ at every step. With a little more cleverness, we can optimize this procedure to only update the entries of the matrix $A$ we need to form the approximation $\hat{A} = FF^*$ . See Algorithm 2 in this preprint of my coauthors and I for details.

How should we choose the pivots? Two simple strategies immediately suggest themselves:

Uniformly random. At each step $j$ , select $s_j$ uniformly at random from among the un-selected pivot indices.
Greedy.⁷ At each step $j$ , select $s_j$ according to the largest diagonal entry of the current residual $A$ :
$s_j = \argmax_{1\le k\le N} a_{kk}.$

The greedy strategy often (but not always) works well, and the uniformly random approach can work surprisingly well if the matrix $A$ is “incoherent”, with all rows and columns of the matrix possessing “similar importance”.

Despite often working fairly well, both the uniform and greedy schemes can fail significantly, producing very low-quality approximations. My research (joint with Yifan Chen, Joel A. Tropp, and Robert J. Webber) has investigated a third strategy striking a balance between these two approaches:

Diagonally weighted random. At each step $j$ , select $s_j$ at random according to the probability weights based on the current diagonal of the matrix
$\mathbb{P} \{ s_j = k \} = \frac{a_{kk}}{\operatorname{tr} A}.$

Our paper provides theoretical analysis and empirical evidence showing that this diagonally-weighted random pivot selection (which we call randomly pivoted Cholesky aka RPCholesky) performs well at approximating all positive semidefinite matrices $A$ , even those for which uniform random and greedy pivot selection fail. The success of this approach can be seen in the following examples (from Figure 1 in the paper), which shows RPCholesky can produce much smaller errors than the greedy and uniform methods.

Conclusions

In this post, we’ve seen that a column Nyström approximation can be obtained from a partial Cholesky decomposition. The residual of the approximation is the Schur complement. I hope you agree that this is a very nice connection between these three ideas. But beyond its mathematical beauty, why do we care about this connection? Here are my reasons for caring:

Analysis. Cholesky factorization and the Schur complement are very well-studied in matrix theory. We can use known facts about Cholesky factorization and Schur complements to prove things about the Nyström approximation, as we did when we invoked the positivity of the Schur complement.
Algorithms. Cholesky factorization-based algorithms like randomly pivoted Cholesky are effective in practice at producing high-quality column Nyström approximations.

On a broader level, our tale of Nyström, Cholesky, and Schur demonstrates that there are rich connections between low-rank approximation and (partial versions of) classical matrix factorizations like LU (with partial pivoting), Cholesky, QR, eigendecomposition, and SVD for to full-rank matrices. These connections can be vital to analyzing low-rank approximation algorithms and developing improvements.

Low-Rank Approximation Toolbox: Nyström Approximation

October 11, 2022 by Ethan Epperly 1 Comment

Welcome to a new series for this blog, Low-Rank Approximation Toolbox. As I discussed in a previous post, many matrices we encounter in applications are well-approximated by a matrix with a small rank. Efficiently computing low-rank approximations has been a major area of research, with applications in everything from classical problems in computational physics and signal processing to trendy topics like data science. In this series, I want to explore some broadly useful algorithms and theoretical techniques in the field of low-rank approximation.

I want to begin this series by talking about one of the fundamental types of low-rank approximation, the Nyström approximation of a $N\times N$ (real symmetric or complex Hermitian) positive semidefinite (psd) matrix $A$ . Given an arbitrary $N\times k$ “test matrix” $\Omega$ , the Nyström approximation is defined to be

(1) $A\langle \Omega\rangle := A\Omega \, (\Omega^*A\Omega)^{-1} \, \Omega^*A.$

This formula is sensible whenever $\Omega^*A\Omega$ is invertible; if $\Omega^*A\Omega$ is not invertible, then the inverse ${}^{-1}$ should be replaced by the Moore–Penrose pseudoinverse ${}^\dagger$ . For simplicity, I will assume that $\Omega^* A \Omega$ is invertible in this post, though everything we discuss will continue to work if this assumption is dropped. I use ${}^*$ to denote the conjugate transpose of a matrix, which agrees with the ordinary transpose ${}^\top$ for real matrices. I will use the word self-adjoint to refer to a matrix which satisfies $A=A^*$ .

The Nyström approximation (1) answers the question

What is the “best” rank- $k$ approximation to the psd matrx $A$ provided only with the matrix–matrix product $A\Omega$ , where $\Omega$ is a known $N\times k$ matrix ( $k\ll N$ )?

Indeed, if we let $Y = A\Omega$ , we observe that the Nyström approximation can be written entirely using $Y$ and $\Omega$ :

$A\langle \Omega\rangle = Y \, (\Omega^* Y)^{-1}\, Y^*.$

This is central advantage of the Nyström approximation: to compute it, the only access to the matrix $A$ I need is the ability to multiply the matrices $A$ and $\Omega$ . In particular, I only need a single pass over the entries of $A$ to compute the Nyström approximation. This allows the Nyström approximation to be used in settings when other low-rank approximations wouldn’t work, such as when $A$ is streamed to me as a sum of matrices that must be processed as they arrive and then discarded.

Choosing the Test Matrix

Every choice of $N\times k$ test matrix $\Omega$ defines a rank- $k$ Nyström approximation¹ $A\langle \Omega\rangle$ by (1). Unfortunately, the Nyström approximation won’t be a good low-rank approximation for every choice of $\Omega$ . For an example of what can go wrong, if we pick $\Omega$ to have columns selected from the eigenvectors of $A$ with small eigenvalues, the approximation $A\langle \Omega\rangle$ will be quite poor.

The very best choice of $\Omega$ would be the $k$ eigenvectors associated with the largest eigenvalues. Unfortunately, computing the eigenvectors to high accuracy is computationally costly. Fortunately, we can get decent low-rank approximations out of much simpler $\Omega$ ‘s:

Random: Perhaps surprisingly, we get a fairly low-rank approximation out of just choosing $\Omega$ to be a random matrix, say, populated with statistically independent standard normal random entries. Intuitively, a random matrix is likely to have columns with meaningful overlap with the large-eigenvalue eigenvectors of $A$ (and indeed with any $k$ fixed orthonormal vectors). One can also pick more exotic kinds of random matrices, which can have computational benefits.
Random then improve: The more similar the test matrix $\Omega$ is to the large-eigenvalue eigenvectors of $A$ , the better the low-rank approximation will be. Therefore, it makes sense to use the power method (usually called subspace iteration in this context) to improve a random initial test matrix $\Omega_{\rm init}$ to be closer to the eigenvectors of $A$ .² See section 11.6 of the following survey for details.
Column selection: If $\Omega$ consists of columns $i_1,i_2,\ldots,i_k$ of the identity matrix, then $A\Omega$ just consists of columns $i_1,\ldots,i_k$ of $A$ : In MATLAB notation,

$A(:,\{i_1,\ldots,i_k\}) = A\Omega \quad \text{for}\quad \Omega = I(:,\{i_1,i_2,\ldots,i_k\}).$
This is highly appealing as it allows us to approximate the matrix $A$ by only reading a small fraction of its entries (provided $k\ll N$ )! Producing a good low-rank approximation requires selecting the right column indices $i_1,\ldots,i_k$ (usually under the constraint of reading a small number of entries from $A$ ). In my research with Yifan Chen, Joel A. Tropp, and Robert J. Webber, I’ve argued that the most well-rounded algorithm for this task is a randomly pivoted partial Cholesky decomposition.

The Projection Formula

Now that we’ve discussed the choice of test matrix, we shall explore the quality of the Nyström approximation as measured by the size of the residual $A - A\langle \Omega\rangle$ . As a first step, we shall show that the residual is psd. This means that $A\langle \Omega\rangle$ is an underapproximation to $A$ .

The positive semidefiniteness of the residual follows from the following projection formula for the Nyström approximation:

$A\langle \Omega \rangle = A^{1/2} P_{A^{1/2}\Omega} A^{1/2}.$

Here, $P_{A^{1/2}\Omega}$ denotes the the orthogonal projection onto the column space of the matrix $A^{1/2}\Omega$ . To deduce the projection formula, we break down $A$ as $A = A^{1/2}\cdot A^{1/2}$ in (1):

$A\langle \Omega\rangle = A^{1/2} \left( A^{1/2}\Omega \left[ (A^{1/2}\Omega)^* A^{1/2}\Omega \right]^{-1} (A^{1/2}\Omega)^* \right) A^{1/2}.$

The fact that the paranthesized quantity is $P_{A^{1/2}\Omega}$ can be verified in a variety of ways, such as by QR factorization.³

With the projection formula in hand, we easily obtain the following expression for the residual:

$A - A\langle \Omega\rangle = A^{1/2} (I - P_{A^{1/2}\Omega}) A^{1/2}.$

To show that this residual is psd, we make use of the conjugation rule.

Conjugation rule: For a matrix $B$ and a self-adjoint matrix $H$ , if $H$ is psd then $B^*HB$ is psd. If $B$ is invertible, then the converse holds: if $B^*HB$ is psd, then $H$ is psd.

The matrix $I - P_{A^{1/2}\Omega}$ is an orthogonal projection and therefore psd. Thus, by the conjugation rule, the residual of the is Nyström approximation is psd:

$A - A\langle \Omega\rangle = \left(A^{1/2}\right)^* (I-P_{A^{1/2}\Omega})A^{1/2} \quad \text{is psd}.$

Optimality of the Nyström Approximation

There’s a question we’ve been putting off that can’t be deferred any longer:

Is the Nyström approximation actually a good low-rank approximation?

As we discussed earlier, the answer to this question depends on the test matrix $\Omega$ . Different choices for $\Omega$ give different approximation errors. See the following papers for Nyström approximation error bounds with different choices of $\Omega$ . While the Nyström approximation can be better or worse depending on the choice of $\Omega$ , what is true about Nyström approximation for every choice of $\Omega$ is that:

The Nyström approximation $A\langle \Omega\rangle$ is the best possible rank- $k$ approximation to $A$ given the information $A\Omega$ .

In precise terms, I mean the following:

Theorem: Out of all self-adjoint matrices $\hat{A}$ spanned by the columns of $A\Omega$ with a psd residual $A - \hat{A}$ , the Nyström approximation has the smallest error as measured by either the spectral or Frobenius norm (or indeed any unitarily invariant norm, see below).

Let’s break this statement down a bit. This result states that the Nyström approximation is the best approximation $\hat{A}$ to $A$ under three conditions:

$\hat{A}$ is self-adjoint.
$\hat{A}$ is spanned by the columns of $A\Omega$ .

I find these first two requirements to be natural. Since $A$ is self-adjoint, it makes sense to require our approximation $\hat{A}$ to be as well. The stipulation that $\hat{A}$ is spanned by the columns $A\Omega$ seems like a very natural requirement given we want to think of approximations which only use the information $A\Omega$ . Additionally, requirement 2 ensures that $\hat{A}$ has rank at most $k$ , so we are really only considering low-rank approximations to $A$ .

The last requirement is less natural:

The residual $A - \hat{A}$ is psd.

This is not an obvious requirement to impose on our approximation. Indeed, it was a nontrivial calculation using the projection formula to show that the Nyström approximation itself satisfies this requirement! Nevertheless, this third stipulation is required to make the theorem true. The Nyström approximation (1) is the best “underapproximation” to the matrix $A$ to in the span of $A\Omega$ .

Intermezzo: Unitarily Invariant Norms and the Psd Order

To prove our theorem about the optimality of the Nyström approximation, we shall need two ideas from matrix theory: unitarily invariant norms and the psd order. We shall briefly describe each in turn.

A norm $\left\|\cdot\right\|_{\rm UI}$ defined on the set of $N\times N$ matrices is said to be unitarily invariant if the norm of a matrix does not change upon left- or right-multiplication by a unitary matrix:

$\left\|UBV\right\|_{\rm UI} = \left\|B\right\|_{\rm UI} \quad \text{for all unitary matrices $U$ and $V$.}$

Recall that a unitary matrix $U$ (called a real orthogonal matrix if $U$ is real-valued) is one that obeys $U^*U = UU^* = I$ . Unitary matrices preserve the Euclidean lengths of vectors, which makes the class of unitarily invariant norms highly natural. Important examples include the spectral, Frobenius, and nuclear matrix norms:

The unitarily invariant norm of a matrix $B$ depends entirely on its singular values $\sigma(B)$ . For instance, the spectral, Frobenius, and nuclear norms take the forms

$\begin{align*}\left\|B\right\|_{\rm op} &= \sigma_1(B),& &\text{(spectral)} \\\left\|B\right\|_{\rm F} &= \sqrt{\sum_{j=1}^N (\sigma_j(B))^2},& &\text{(Frobenius)} \\\left\|B\right\|_{*} &=\sum_{j=1}^N \sigma_j(B)).& &\text{(nuclear)}\end{align*}$

In addition to being entirely determined by the singular values, unitarily invariant norms are non-decreasing functions of the singular values: If the $j$ th singular value of $B$ is larger than the $j$ th singular value of $C$ for $1\le j\le N$ , then $\left\|B\right\|_{\rm UI}\ge \left\|C\right\|_{\rm UI}$ for every unitarily invariant norm $\left\|\cdot\right\|_{\rm UI}$ . For more on unitarily invariant norms, see this short and information-packed blog post from Nick Higham.

Our second ingredient is the psd order (also known as the Loewner order). A self-adjoint matrix $A$ is larger than a self-adjoint matrix $H$ according to the psd order, written $A\succeq H$ , if the difference $A-H$ is psd. As a consequence, $A\succeq 0$ if and only if $A$ is psd, where $0$ here denotes the zero matrix of the same size as $A$ . Using the psd order, the positive semidefiniteness of the Nyström residual can be written as $A - A\langle \Omega\rangle \succeq 0$ .

If $A$ and $H$ are both psd matrices and $A$ is larger than $H$ in the psd order, $A\succeq H\succeq 0$ , it seems natural to expect that $A$ is larger than $H$ in norm. Indeed, this intuitive statement is true, at least when one restricts oneself to unitarily invariant norms.

Psd order and norms. If $A\succeq H\succeq 0$ , then $\left\|A\right\|_{\rm UI} \ge \left\|H\right\|_{\rm UI}$ for every unitarily invariant norm $\left\|\cdot\right\|_{\rm UI}$ .

This fact is a consequence of the following observations:

If $A\succeq H$ , then the eigenvalues of $A$ are larger than $H$ in the sense that the $j$ th largest eigenvalue of $A$ is larger than the $j$ th largest eigenvalue of $H$ .
The singular values of a psd matrix are its eigenvalues.
Unitarily invariant norms are non-decreasing functions of the singular values.

Optimality of the Nyström Approximation: Proof

In this section, we’ll prove our theorem showing the Nyström approximation is the best low-rank approximation satisfying properties 1, 2, and 3. To this end, let $\hat{A}$ be any matrix satisfying properties 1, 2, and 3. Because of properties 1 (self-adjointness) and 2 (spanned by columns of $A\Omega$ ), $\hat{A}$ can be written in the form

$\hat{A} = A\Omega \, T \, (A\Omega)^* = A \Omega \, T \, \Omega^*A,$

where $T$ is a self-adjoint matrix. To make this more similar to the projection formula, we can factor $A^{1/2}$ on both sides to obtain

$\hat{A} = A^{1/2} (A^{1/2}\Omega\, T\, \Omega^*A^{1/2}) A^{1/2}.$

To make this more comparable to the projection formula, we can reparametrize by introducing a matrix $Q$ with orthonormal columns with the same column space as $A^{1/2}\Omega$ . Under this parametrization, $\hat{A}$ takes the form

$\hat{A} = A^{1/2} \,QMQ^*\, A^{1/2} \quad \text{where} \quad M\text{ is self-adjoint}.$

The residual of this approximation is

(2) $A - \hat{A} = A^{1/2} (I - QMQ^*)A^{1/2}.$

We now make use of the of conjugation rule again. To simplify things, we make the assumption that $A$ is invertible. (As an exercise, see if you can adapt this argument to the case when this assumption doesn’t hold!) Since $A - \hat{A}\succeq 0$ is psd (property 3), the conjugation rule tells us that

$I - QMQ^*\succeq 0.$

What does this observation tell us about $M$ ? We can apply the conjugation rule again to conclude

$Q^*(I - QMQ^*)Q = Q^*Q - (Q^*Q)M(Q^*Q) = I-M \succeq 0.$

(Notice that $Q^*Q = I$ since $Q$ has orthonormal columns.)

We are now in a place to show that $A - \hat{A}\succeq A - A\langle \Omega\rangle$ . Indeed,

$\begin{align*}A - \hat{A} - (A-A\langle \Omega\rangle) &= A\langle\Omega\rangle - \hat{A} \\&= A^{1/2}\underbrace{QQ^*}_{=P_{A^{1/2}\Omega}}A^{1/2} - A^{1/2}QMQ^*A^{1/2} \\&=A^{1/2}Q(I-M)Q^*A^{1/2}\\&\succeq 0.\end{align*}$

The second line is the projection formula together with the observation that $P_{A^{1/2\Omega}} = QQ^*$ and the last line is the conjugation rule combined with the fact that $I-M$ is psd. Thus, having shown

$A - \hat{A} \succeq A - A\langle\Omega\rangle \succeq 0,$

we conclude

$\|A - \hat{A}\|_{\rm UI} \ge \left\|A - A\langle \Omega\rangle\right\|_{\rm UI} \quad \text{for every unitarily invariant norm $\left\|\cdot\right\|_{\rm UI}$}.$