Statistical Query Algorithms and Low-Degree Tests Are Almost Equivalent

by   Matthew Brennan, et al.

Researchers currently use a number of approaches to predict and substantiate information-computation gaps in high-dimensional statistical estimation problems. A prominent approach is to characterize the limits of restricted models of computation, which on the one hand yields strong computational lower bounds for powerful classes of algorithms and on the other hand helps guide the development of efficient algorithms. In this paper, we study two of the most popular restricted computational models, the statistical query framework and low-degree polynomials, in the context of high-dimensional hypothesis testing. Our main result is that under mild conditions on the testing problem, the two classes of algorithms are essentially equivalent in power. As corollaries, we obtain new statistical query lower bounds for sparse PCA, tensor PCA and several variants of the planted clique problem.



page 1

page 2

page 3

page 4


Reducibility and Computational Lower Bounds for Problems with Planted Sparse Structure

The prototypical high-dimensional statistics problem entails finding a s...

Sharp Computational-Statistical Phase Transitions via Oracle Computational Model

We study the fundamental tradeoffs between computational tractability an...

The Franz-Parisi Criterion and Computational Trade-offs in High Dimensional Statistics

Many high-dimensional statistical inference problems are believed to pos...

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

These notes survey and explore an emerging method, which we call the low...

Phase Transitions for High Dimensional Clustering and Related Problems

Consider a two-class clustering problem where we observe X_i = ℓ_i μ + Z...

Property Testing in High Dimensional Ising models

This paper explores the information-theoretic limitations of graph prope...

Statistical Limits of Convex Relaxations

Many high dimensional sparse learning problems are formulated as nonconv...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.