PTF Testing Lower Bounds for Non-Gaussian Component Analysis

TL;DR

This paper establishes the first non-trivial lower bounds for Polynomial Threshold Function (PTF) tests in statistical problems, specifically proving near-optimal bounds for Non-Gaussian Component Analysis, highlighting fundamental computational limits.

Contribution

It provides the first non-trivial PTF testing lower bounds for NGCA and related problems, advancing understanding of computational hardness in statistical testing.

Findings

Proved near-optimal PTF testing lower bounds for NGCA

Developed new structural tools for analyzing low-degree polynomials

Connected PTF lower bounds to pseudorandom generator techniques

Abstract

This work studies information-computation gaps for statistical problems. A common approach for providing evidence of such gaps is to show sample complexity lower bounds (that are stronger than the information-theoretic optimum) against natural models of computation. A popular such model in the literature is the family of low-degree polynomial tests. While these tests are defined in such a way that make them easy to analyze, the class of algorithms that they rule out is somewhat restricted. An important goal in this context has been to obtain lower bounds against the stronger and more natural class of low-degree Polynomial Threshold Function (PTF) tests, i.e., any test that can be expressed as comparing some low-degree polynomial of the data to a threshold. Proving lower bounds against PTF tests has turned out to be challenging. Indeed, we are not aware of any non-trivial PTF testing lower bounds in the literature. In this paper, we establish the first non-trivial PTF testing lower bounds for a range of statistical tasks. Specifically, we prove a near-optimal PTF testing lower bound for Non-Gaussian Component Analysis (NGCA). Our NGCA lower bound implies similar lower bounds for a number of other statistical problems. Our proof leverages a connection to recent work on pseudorandom generators for PTFs and recent techniques developed in that context. At the technical level, we develop several tools of independent interest, including novel structural results for analyzing the behavior of low-degree polynomials restricted to random directions.