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

• Published in: arXiv.org 2024-03
• Main Authors: Diakonikolas, Ilias, Kane, Daniel, Ren, Lisheng, Sun, Yuxin
• Format: Article
• Language: English
• Subjects: Chi-square test; Lower bounds; Matching; Normal distribution
• ISSN: 2331-8422

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.