*This post is part of a new series for this blog, Note to Self, where I collect together some notes about an idea related to my research. This content may be much more technical than most of the content of this blog and of much less wide interest. My hope in sharing this is that someone will find this interesting and useful for their own work.*

This post is about a fundamental tool of high-dimensional probability, the Hanson–Wright inequality. The Hanson–Wright inequality is a concentration inequality for *quadratic forms* of random vectors—that is, expressions of the form where is a random vector. Many statements of this inequality in the literature have an unspecified constant ; our goal in this post will be to derive a fairly general version of the inequality with only explicit constants.

The core object of the Hanson–Wright inequality is a *subgaussian* random variable. A random variable is subgaussian if the probability it exceeds a threshold in magnitude decays as

(1)

The name sub*gaussian* is appropriate as the tail probabilities of Gaussian random variables exhibit the same square-exponential decrease .

A (non-obvious) fact is that if is subgaussian in the sense (1) and centered (), then ‘s *cumulant generating function* (cgf)

is subquadratic: There is a constant (independent of and ), for which

(2)

Moreover,^{1}See Proposition 2.5.2 of Vershynin’s *High-Dimensional Probability*. a subquadratic cgf (2) also implies the subgaussian tail property (1), with a different parameter .

Since properties (1) and (2) are equivalent (up to a change in the parameter ), we are free to fix a version of property (2) as our definition for a (centered) subgaussian random variable.

Definition (subgaussian random variable):A centered random variable is said to be-subgaussianorsubgaussian with variance proxyif its cgf is subquadratic:(3)

For instance, a mean-zero Gaussian random variable with variance has cgf

(4)

and is thus subgaussian with variance proxy equal to its variance.

Here is a statement of the Hanson–Wright inequality as it typically appears with unspecified constants (see Theorem 6.2.1 of Vershynin’s *High-Dimensional Probability*):

Theorem (Hanson–Wright):Let be a random vector with independent centered -subgaussian entries and let be a square matrix. Thenwhere is a constant (not depending on , , , or ).

^{2}Here, and denote the Frobenius and spectral norms.

This type of concentration is exactly the same type as provided by Bernstein’s inequality (which I discussed in my post on concentration inequalities). In particular, for small deviations , the tail probabilities decay are *subgaussian* with variance proxy :

For large deviations , this switches to *subexponential* tail probabilities with decay rate :

Mediating these two parameter regimes are the size of the matrix , as measured by its Frobenius and spectral norms, and the degree of subgaussianity of , measured by the variance proxy .

## Diagonal-Free Hanson–Wright

Now we come to a first version of the Hanson–Wright inequality with explicit constants, first for a matrix which is diagonal-free—that is, having all zeros on the diagonal. I obtained this version of the inequality myself, though I am very sure that this version of the inequality or an improvement thereof appears somewhere in the literature.

Theorem (Hanson–Wright, explicit constants, diagonal-free):Let random vector with independent centered -subguassian entries and let be adiagonal-freesquare matrix. Then we have the cgf boundAs a consequence, we have the concentration bound

Let us begin proving this result. Our proof will follow the same steps as Vershynin’s proof in *High-Dimensional Probability* (which in turn is adapted from an article by Rudelson and Vershynin), but taking care to get explicit constants. Unfortunately, proving all of the relevant tools from first principles would easily triple the length of this post, so I make frequent use of results from the literature.

We begin by the *decoupling bound* (Theorem 6.1.1 in Vershynin’s *High-Dimensional Probability*), which allows us to replace one with an independent copy at the cost of a factor of four:

(5)

We seek to compare the bilinear form to the Gaussian bilinear form where and are independent standard Gaussian vectors. We begin with the following cgf bound for the Gaussian quadratic form :

This equation is the result of Example 2.12 in Boucheron, Lugosi, and Massart’s *Concentration Inequalities*. By applying this result to the Hermitian dilation of in ‘s place, one obtains a similar result for the decoupled bilinear form :

(6)

We now seek to compare to . To do this, we first evaluate the cgf of only over the randomness in . Since we’re only taking an expectation over the random variable , we can apply the subquadratic tail condition (3) to obtain

(7)

Now we perform a similar computation for the quantity in which has been replaced by the Gaussian vector :

We stress that this is an *equality* since the cgf of a Gaussian random variable is given by (4). Thus we can substitute the left-hand side of the above display into the right-hand side of (7), yielding

(8)

We now perform this same trick again using the randomness in :

(9)

Packaging up (8) and (9) gives

(10)

Combining all these results (5), (6), and (10), we obtain

This cgf implies the desired probability bound as a consequence of the following fact (see Boucheron, Lugosi, and Massart’s *Concentration Inequalities* page 29 and Exercise 2.8):

Fact (Bernstein concentration from Bernstein cgf bound):Suppose that a random variable satisfies the cgf bound for . Then

## General Hanson–Wright

Now, here’s a more general result (with worse constants) which permits the matrix to possess a diagonal.

Theorem (Hanson–Wright, explicit constants):Let random vector with independent centered -subguassian entries and let be anarbitrarysquare matrix. Then we have the cgf boundAs a consequence, we have the concentration bound

Decompose the matrix into its diagonal and off-diagonal portions. For any two random variables and (possibly highly dependent), we can bound the cgf of their sum using the following “union bound”:

(11)

The two equality statements are the definition of the cumulant generating function and the inequality is Cauchy–Schwarz.

Using the “union bound”, it is sufficient to obtain bounds for the cgfs of the diagonal and off-diagonal parts and . We begin with the diagonal part. We compute

(12)

For the cgf of , we use the following bound, taken from Appendix B of the following paper:

Substituting this result into (12) gives

(13)

For the second inequality, we used the facts that and .

We now look at the off-diagonal part . We use a version of the decoupling bound (5) where we compare to , where we’ve both replaced one copy of with an independent copy and reinstated the diagonal of (see Remark 6.1.3 in Vershynin’s *High-Dimensional Probability*):

We can now just repeat the rest of the argument for the diagonal-free Hanson–Wright inequality, yielding the same conclusion

(14)

Combining (11), (13), and (14), we obtain

As with above, this cgf bound implies the desired probability bound.