Notes on Computational Hardness of Hypothesis Testing: Predictions Using the Low-Degree Likelihood Ratio

Abstract

We study when low coordinate degree functions (LCDF)-linear combinations of functions depending on small subsets of entries of a vector-can hypothesis test between high-dimensional probability measures. These functions are a generalization, proposed in Hopkins' 2018 thesis but seldom studied since, of low degree polynomials (LDP), a class widely used in recent literature as a proxy for all efficient algorithms for tasks in statistics and optimization. Instead of the orthogonal polynomial decompositions used in LDP calculations, our analysis of LCDF is based on the Efron-Stein or ANOVA decomposition, making it much more broadly applicable. By way of illustration, we prove channel universality for the success of LCDF in testing for the presence of sufficiently "dilute" random signals through noisy channels: the efficacy of LCDF depends on the channel only through the scalar Fisher information for a class of channels including nearly arbitrary additive i.i.d. noise and nearly arbitrary exponential families. As applications, we extend lower bounds against LDP for spiked matrix and tensor models under additive Gaussian noise to lower bounds against LCDF under general noisy channels. We also give a simple and unified treatment of the effect of censoring models by erasing observations at random and of quantizing models by taking the sign of the observations. These results are the first computational lower bounds against any large class of algorithms for all of these models when the channel is not one of a few special cases, and thereby give the first substantial evidence for the universality of several statistical-to-computational gaps.

Statistical limits of correlation detection in trees

“…These quantities have been proved to remain finite when λ → ∞ by property (15) of Theorem 4. This shows that all the cumulants of (y, y ′ ) of order ≥ 3 tend to 0 when λ → ∞, and hence establishes the desired convergence in distribution.…”

“…Remark 3.1. A hint of the Gaussian convergence in the limit λ → ∞ can be a posteriori read from Equation (15). To explain this point let us first define the Hermite polynomials…”

“…, x (n) ) contributes are in finite number (independent of λ) and are all of order O(λ −1/2 ). Taking λ → ∞ thus establishes property (15) for the rank d + 1. Here again, previous computations are made rigorous since the trees are finite, by noticing that the series in (20) has infinite radius of convergence, and appealing to Fubini's theorem.…”

“…Remark 2.1. Note that the right hand side of (15) simplifies for n = 2 to 1 β (1) =β (2) ; (13) shows that for n = 2 the equality in (15) holds for all λ, not only in the limit λ → ∞, which is nevertheless required for n > 2. Note also that (12) combined with (13) implies the following first moment condition:…”

“…The decomposition (11) may look reminiscent of the low-degree polynomials approach to hypothesis testing problems (see e.g. [15,16]), in which the likelihood ratio is projected onto the subspace of restricted degree polynomials in the observations. This similarity hides some differences, in particular (11) provides an exact expression of the likelihood ratio and not an approximation, and also the observations are here infinite-dimensional for d ≥ 2, which makes the expansion basis less immediately apparent than for, say, the entries of the adjacency matrix of a graph.…”