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

• URL: http://arxiv.org/abs/2403.04744v1
• Date: Thu, 7 Mar 2024 18:49:32 GMT
• Title: SQ Lower Bounds for Non-Gaussian Component Analysis with Weaker Assumptions
• Authors: Ilias Diakonikolas, Daniel Kane, Lisheng Ren and Yuxin Sun
• Abstract summary: We study the complexity of Non-Gaussian Component Analysis (NGCA) in the Statistical Query model. We prove near-optimal SQ lower bounds for NGCA under the moment-matching condition only.
• Score: 50.20087216230159
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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