The purpose of this note is to describe the Gaussian hypercontractivity inequality. As an application, we’ll obtain a weaker version of the Hanson–Wright inequality.
The Noise Operator
We begin our discussion with the following question:
Let be a function. What happens to , on average, if we perturb its inputs by a small amount of Gaussian noise?
Let’s be more specific about our noise model. Let be an input to the function and fix a parameter (think of as close to 1). We’ll define the noise corruption of to be
Here, is the standard multivariate Gaussian distribution. In our definition of , we both add Gaussian noise and shrink the vector by a factor . In particular, we highlight two extreme cases:
- No noise. If , then there is no noise and .
- All noise. If , then there is all noise and . The influence of the original vector has been washed away completely.
The noise corruption (1) immediately gives rise to the noise operator1The noise operator is often called the Hermite operator. The noise operator is related to the Ornstein–Uhlenbeck semigroup operator by a change of variables, . . Let be a function. The noise operator is defined to be:
The noise operator computes the average value of when evaluated at the noisy input . Observe that the noise operator maps a function to another function . Going forward, we will write to denote .
To understand how the noise operator acts on a function , we can write the expectation in the definition (2) as an integral:
Here, denotes the (Euclidean) length of . We see that is the convolution of with a Gaussian density. Thus, acts to smooth the function .
See below for an illustration. The red solid curve is a function , and the blue dashed curve is .
As we decrease from to , the function is smoothed more and more. When we finally reach , has been smoothed all the way into a constant.
The noise operator converts a function to another function . We can evaluate these two functions at a Gaussian random vector , resulting in two random variables and .
We can think of as a modification of the random variable where “a fraction of the variance of has been averaged out”. We again highlight the two extreme cases:
- No noise. If , . None of the variance of has been averaged out.
- All noise. If , is a constant random variable. All of the variance of has been averaged out.
Just as decreasing smoothes the function until it reaches a constant function at , decreasing makes the random variable more and more “well-behaved” until it becomes a constant random variable at . This “well-behavingness” property of the noise operator is made precise by the Gaussian hypercontractivity theorem.
Moments and Tails
In order to describe the “well-behavingness” properties of the noise operator, we must answer the question:
How can we measure how well-behaved a random variable is?
There are many answers to this question. For this post, we will quantify the well-behavedness of a random variable by using the norm.2Using norms is a common way of measuring the niceness of a function or random variable in applied math. For instance, we can use Sobolev norms or reproducing kernel Hilbert space norms to measure the smoothness of a function in approximation theory, as I’ve discussed before on this blog.
The norm of a (-valued) random variable is defined to be
(3)The th power of the norm is sometimes known as the th absolute moment of .
The norms of random variables control the tails of a random variable—that is, the probability that a random variable is large in magnitude. A random variables with small tails is typically thought of as a “nice” or “well-behaved” random variable. Random quantities with small tails are usually desirable in applications, as they are more predictable—unlikely to take large values.
The connection between tails and norms can be derived as follows. First, write the tail probability for using th powers:
Then, we apply Markov’s inequality, obtaining
(4)We conclude that a random variable with finite norm (i.e., ) has tails that decay at at a rate or faster.
Before we introduce the Gaussian hypercontractivity theorem, let’s establish a weaker property of the noise operator, contractivity.
Proposition 1 (Gaussian contractivity). Choose a noise level and a power , and let be a Gaussian random vector. Then contracts the norm of :
This result shows that the noise operator makes the random variable no less nice than was.
Gaussian contractivity is easy to prove. Begin using the definition of the noise operator (2) and norm (3):
Now, we can apply Jensen’s inequality to the convex function , obtaining
Finally, realize that for the independent normal random vectors, we have
Thus, has the same distribution as . Thus, using in place of , we obtain
Gaussian contractivity (Proposition 1) is proven.
The Gaussian contractivity theorem shows that is no less well-behaved than is. In fact, is more well-behaved than is. This is the content of the Gaussian hypercontractivity theorem:
Theorem 2 (Gaussian hypercontractivity): Choose a noise level and a power , and let be a Gaussian random vector. Then
In particular, for ,
We have highlighted the case because it is the most useful in practice.
This result shows that as we take smaller, the random variable becomes more and more well-behaved, with tails decreasing at a rate
The rate of tail decrease becomes faster and faster as becomes closer to zero.
We will prove the Gaussian hypercontractivity at the bottom of this post. For now, we will focus on applying this result.
A multilinear polynomial is a multivariate polynomial in the variables in which none of the variables is raised to a power higher than one. So,
(5)is multilinear, but
is not multilinear (since is squared).
For multilinear polynomials, we have the following very powerful corollary of Gaussian hypercontractivity:
Corollary 3 (Absolute moments of a multilinear polynomial of Gaussians). Let be a multilinear polynomial of degree . (That is, at most variables occur in any monomial of .) Then, for a Gaussian random vector and for all ,
Let’s prove this corollary. The first observation is that the noise operator has a particularly convenient form when applied to a multilinear polynomial. Let’s test it out on our example (5) from above. For
We see that the expectation applies to each variable separately, resulting in each replaced by . This trend holds in general:
Proposition 4 (noise operator on multilinear polynomials). For any multilinear polynomial , .
We can use Proposition 4 to obtain bounds on the norms of multilinear polynomials of a Gaussian random variable. Indeed, observe that
Thus, by Gaussian hypercontractivity, we have
The final step of our argument will be to compute . Write as
Since is multilinear, for . Since is degree-, . The multilinear monomials are orthonormal with respect to the inner product:
(See if you can see why!) Thus, by the Pythagorean theorem, we have
Similarly, the coefficients of are . Thus,
Thus, putting all of the ingredients together, we have
Setting (equivalently ), Corollary 3 follows.
To see the power of the machinery we have developed, let’s prove a version of the Hanson–Wright inequality.
Theorem 5 (suboptimal Hanson–Wright). Let be a symmetric matrix with zero on its diagonal and be a Gaussian random vector. Then
Hanson–Wright has all sorts of applications in computational mathematics and data science. One direct application is to obtain probabilistic error bounds for the error incurred by a stochastic trace estimation formulas.
This version of Hanson–Wright is not perfect. In particular, it does not capture the Bernstein-type tail behavior of the classical Hanson–Wright inequality
But our suboptimal Hanson–Wright inequality is still pretty good, and it requires essentially no work to prove using the hypercontractivity machinery. The hypercontractivity technique also generalizes to settings where some of the proofs of Hanson–Wright fail, such as multilinear polynomials of degree higher than two.
Let’s prove our suboptimal Hanson–Wright inequality. Set . Since has zero on its diagonal, is a multilinear polynomial of degree two in the entries of . The random variable is mean-zero, and a short calculation shows its norm is
Thus, by Corollary 3,
(6)In fact, since the norms are monotone, (6) holds for as well. Therefore, the standard tail bound for norms (4) gives
Now, we must optimize the value of to obtain the sharpest possible bound. To make this optimization more convenient, introduce a parameter
In terms of the parameter, the bound (7) reads
The tail bound is minimized by taking , yielding the claimed result
Proof of Gaussian Hypercontractivity
Let’s prove the Gaussian hypercontractivity theorem. For simplicity, we will stick with the case, but the higher-dimensional generalizations follow along similar lines. The key ingredient will be the Gaussian Jensen inequality, which made a prominent appearance in a previous blog post of mine. Here, we will only need the following version:
(8)holds for all test functions if, and only if,
To me, this proof of Gaussian hypercontractivity using Gaussian Jensen (adapted from Paata Ivanishvili‘s excellent post) is amazing. First, we reformulate the Gaussian hypercontractivity property a couple of times using some functional analysis tricks. Then we do a short calculation, invoke Gaussian Jensen, and the theorem is proved, almost as if by magic.
Part 1: Tricks
Let’s begin with “tricks” part of the argument.
Trick 1. To prove Gaussian hypercontractivity holds for all functions , it is sufficient to prove for all nonnegative functions .
Indeed, suppose Gaussian hypercontractivity holds for all nonnegative functions . Then, for any function , apply Jensen’s inequality to conclude
Thus, assuming hypercontractivity holds for the nonnegative function , we have
Thus, the conclusion of the hypercontractivity theorem holds for as well, and the Trick 1 is proven.
Trick 2. To prove Gaussian hypercontractivity for all , it is sufficient to prove the following “bilinearized” Gaussian hypercontractivity result:
holds for all with . Here, is the Hölder conjugate to .
Indeed, this follows3This argument may be more clear to parse if we view and as functions on equipped with the standard Gaussian measure . This result is just duality for the norm. from the dual characterization of the norm of :
Trick 2 is proven.
Trick 3. Let be a pair of standard Gaussian random variables with correlation . Then the bilinearized Gaussian hypercontractivity statement is equivalent to
Indeed, define for the random variable in the definition of the noise operator . The random variable is standard Gaussian and has correlation with , concluding the proof of Trick 3.
Finally, we apply a change of variables as our last trick:
Trick 4. Make the change of variables and , yielding the final equivalent version of Gaussian hypercontractivity:
for all functions and (in the appropriate spaces).
Part 2: Calculation
We recognize this fourth equivalent version of Gaussian hypercontractivity as the conclusion (8) to Gaussian Jensen with
. Thus, to prove Gaussian hypercontractivity, we just need to check the hypothesis (9) of the Gaussian Jensen inequality (Theorem 6).
We now enter the calculation part of the proof. First, we compute the Hessian of :
We have written for the Hölder conjugate to . By Gaussian Jensen, to prove Gaussian hypercontractivity, it suffices to show that
is negative semidefinite for all . There are a few ways we can make our lives easier. Write this matrix as
Scaling by nonnegative and conjugation both preserve negative semidefiniteness, so it is sufficient to prove
Since the diagonal entries of are negative, at least one of ‘s eigenvalues is negative.4Indeed, by the Rayleigh–Ritz variational principle, the smallest eigenvalue of a symmetric matrix is Taking for to be each of the standard basis vectors, shows that the smallest eigenvalue of is smaller than the smallest diagonal entry of . Therefore, to prove is negative semidefinite, we can prove that its determinant (= product of its eigenvalues) is nonnegative. We compute
Now, just plug in the values for , , :
Thus, . We conclude is negative semidefinite, proving the Gaussian hypercontractivity theorem.