SQ Lower Bounds for Non-Gaussian Component Analysis with Weaker Assumptions

TL;DR

This paper proves that Non-Gaussian Component Analysis (NGCA) remains computationally hard in the Statistical Query model under weaker assumptions than previously known, specifically removing the need for the chi-squared norm condition.

Contribution

It establishes near-optimal SQ lower bounds for NGCA using only the moment-matching condition, broadening the understanding of its computational complexity.

Findings

Proves SQ-hardness of NGCA without chi-squared norm condition

Generalizes SQ lower bounds to hidden subspace settings

Provides near-optimal bounds for various estimation tasks

Abstract

We study the complexity of Non-Gaussian Component Analysis (NGCA) in the Statistical Query (SQ) model. Prior work developed a general methodology to prove SQ lower bounds for this task that have been applicable to a wide range of contexts. In particular, it was known that for any univariate distribution $A$ satisfying certain conditions, distinguishing between a standard multivariate Gaussian and a distribution that behaves like $A$ in a random hidden direction and like a standard Gaussian in the orthogonal complement, is SQ-hard. The required conditions were that (1) $A$ matches many low-order moments with the standard univariate Gaussian, and (2) the chi-squared norm of $A$ with respect to the standard Gaussian is finite. While the moment-matching condition is necessary for hardness, the chi-squared condition was only required for technical reasons. In this work, we establish that the latter condition is indeed not necessary. In particular, we prove near-optimal SQ lower bounds for NGCA under the moment-matching condition only. Our result naturally generalizes to the setting of a hidden subspace. Leveraging our general SQ lower bound, we obtain near-optimal SQ lower bounds for a range of concrete estimation tasks where existing techniques provide sub-optimal or even vacuous guarantees.