1 Introduction and Background
being the eigenvalues of a symmetric matrix; for example, a quadratic form where and represents applied to the eigenvalues of . As described in Christ (2017) and Christ et al. (2020), results of this kind can be generalized to cases where is asymmetric with careful treatment of .
arises as the limiting distribution of test statistics used in a wide range of applications. These statistics include the Hilbert-Schmidt Information Criterion used for high-dimensional independence testingGretton et al. (2005); Zhang et al. (2018)
, score statistics for linear and genearlized linear mixed models commonly used in genomicsLin (1997); Wu et al. (2011), and the goodness-of-fit statistic proposed by Peña and Rodríguez (2002) for ARMA models in time series analysis. It is easy to see that has mean
Work in Christ et al. (2020) established a concentration inequality to bound the tails of , which yield a set of bounds for different functions. The results of Christ et al. (2020) show that it is possible to find polynomial bounds, but these are not constructed explicitly. We provide here explicit optimal coefficients for bounds of this form in the single-spectrum case. This earlier work yielded the following bound on (by which we designate the base version of , where is the identity function):
Theorem 1 (see p.75 in Christ (2017)).
Let and be a real, symmetric matrix. Let . Let and let , where are the eigenvalues of . Then, for all ,
Similarly, for all ,
The proof of this result relies on a Chernoff-style bound involving the cumulant generating function (cgf) of , which has two main types of terms:
Each of these is bounded by a quadratic function, leading to an overall bound in terms of easily computable coefficients. We improve on this previous work by constructing a family of quadratics that yield pointwise tighter bounds on and . We then show how these can be incorporated into an optimisation step to yield tighter bounds on the tails of .
In Section 2 we present our main results. First we present Lemma 2, which tightens the quadratic bounds above from Christ (2017). From this lemma, we derive the corresponding improved bounds on the tails of in Theorem 3. Specialisation of these results to some particular functions then follow in corollaries. In Section 3, we empirically demonstrate the improvement provided by these bounds with an application to a simulatd matrix with a exponentially decaying spectrum. Section 4 concludes with discussion of potential future improvements. Proofs for the main results are presented in Section 5.
2 Main Results
Our results depend upon elementary upper bounds on and in the form of parabolas passing through the origin. We describe the coefficients of these parabolas in terms of the width of the (symmetric) interval on which the bounds are to be applied, and on the parameter that arises from the cgf. We exploit two openings for improvement: optimising the coefficients of the parabola and optimising the width of the scaled domain over which it bounds and .
Let be a monotonic increasing function such that . Let be a fixed positive real number, and , where
Furthermore, suppose that over the region the following inequalities are satisfied for both and :
For each define
Then for each , among all quadratic function that maintain over the whole region , where
the difference is minimised at every point by the choice and ; and among those that maintain over the whole region , where
the difference is minimised at every point by the choice and .
This lemma will allow us to build on the existing result from Christ (2017). In the original form of this theorem was restricted so that , avoiding the asymptote at . We remove this boundary at 1/4 and allow to get arbitrarily close to . We also reinterpret , so that it now defines the domain of rather than that of . It also means that for every endpoint along the interval we can obtain optimal coefficients on our quadratic bounds. This yields a new bound on the tails of as follows.
Let where , and let be set to . Suppose satisfies the conditions in Lemma 2. Then for all ,
where , and
Furthermore, for all ,
In the central use case, where arises as , we can apply Theorem 3, where and
This allows us to quickly compute tight tail bounds on . In the following corollaries we address special cases of .
while the two conditions for become
All of these inequalities hold for , and so Lemma 1 holds where is the identity function. The result follows by application of Theorem 1. ∎
Since , the from Lemma 1 is equal to . We introduce the variable and note that our original region, and , corresponds
Substituting the definitions of into condition (1) yields
The condition (2) is trivial for even
, while for oddit becomes
For any positive integer , for each
where is defined in (4) and , .
Here we compare the bounds in Corollary 4 and Corollary 6 to the bounds provided in Christ (2017) and Christ et al. (2020) for different matrix powers . For this comparison, we simluated a matrix with an exponentially decaying spectrum of eigenvalues, a case which is relatively common in applications. See Figure 3.1.
For this comparison, we simluated a matrix with an exponentially decaying spectrum of eigenvalues, a case which is relatively common in applications. See Figure 3.1. Note that we have plotted the logarithm (base 10) of the true probability on the axis, and the error in the bounds on the axis. Thus, using the solid red line in Figure 3.1, if the true tail probability of is (), then our new bound for would be approximately of the order .
Particularly of note is that while our bounds show an improvement for all functions satisfying the assumptions of Lemma 2, the improvement is much greater for even functions. This is because our bounds are quadratic, so they must yield the same error bound on both sides of the real line for even functions; however, when bounding an odd function, our bounds will be tight by construction for but may be much looser for . As expected, our bounds perform worse for higher powers , which is effectively a result of attempting to control the higher-order behavior of the matrix given traces that measure the empirical mean and variance of the matrix elements.
We have placed tighter bounds than were previously available on the tails of . Although our bounds are not available in an explicit form, since we optimise over two parameters that previous results set arbitrarily, our bounds are at least as good, which is seen in practice. We further observe that they tend to be significantly tighter and improve relative to the old bounds as we go further out into the tails.
Although our results do give a significantly tighter bound on the tails of , they only work for a specific class of satisfying the conditions of Lemma 1, which notably excludes functions such as . Future developments could improve on this; one possible way would be to introduce an intercept into our quadratic bounds for and , which would maintain the ease of computability while extending it to a wider range of . A further source of improvement may be achieved by modifying Lemma 2 to account for the asymmetry on vs. . Treating each side of the real line separately could enable one to use both the smallest and largest eigenvalue, rather than just .
Though outside the scope of this paper, it would be possible to achieve similar bounds for sub-Gaussian random variables. This would provide tighter results than currently exist in those cases if the Hanson–Wright inequality argument Rudelson et al. (2013) were reworked in terms of explicit constants.
5 Proofs of Main Results
Proof of Lemma 2
In the special case we simply have that , , , , and are all , so the Lemma clearly holds. We assume now .
Since , the choice of is fixed by the need to make 0 a critical point for both of these functions. It remains only to consider the choice of .
Consider being either or . Write , where . Since is fixed, the quadratic functions are strictly increasing in at every point. For define
Then is the minimum such that , and the optimum that we are looking for is . We have
by assumption 1. Thus is non-decreasing in , and so has its maximum at . This shows that taking makes for any , and it is the smallest such . Note that when , and when .
Assumption 2 tells us that for we have . This implies that , so the same choice of provides a bound — that is, — over the whole interval .∎
Proof of Theorem 3
We credit Pollard (2015) for the proof technique used below.
Using Lemma 3.1.3 in (Christ, 2017, p.75), for
By Lemma 2 we know, setting , that for ,
We claim that this is the optimal choice of . Smaller will void the inequalities for some and so cannot be considered. On the other hand, we know that both and are increasing in so any larger would simultaneously weaken the quadratic bound and shrink the range of values to which it can be applied, since is decreasing in .
Applying the definitions of and we have
By Markov’s Inequality, for any ,
For , since is positive we have the trivial bound .
The bound for is derived identically.
The first two authors would like to acknowledge the support of the Engineering and Physical Sciences Research Council [grant number EP/M507854/1]. The last author would like to acknowledge the support of the Summer Opportunities Abroad Program (SOAP) - WUSM Global Health & Medicine and the WUSM Dean’s Fellowship, both from the Washington University School of Medicine in St. Louis.
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