A Correlation-Gap Bound for Nonlinear Gaussian PCA

2026-07-16

AI Summary

This work investigates the optimality of the Karhunen–Loève (KL) basis in nonlinear Gaussian PCA under adaptive coordinate selection, thereby verifying the Mallat–Zeitouni conjecture. By reformulating the problem as an analysis of the correlation gap of rank-$d$ uniform matroids over Gaussian level sets and leveraging the Schur–Horn majorization inequality to relax arbitrary orthogonal transformations, the study establishes, for the first time, a dimension-free approximation guarantee that depends solely on the number of retained coordinates and ensures energy preservation. The results demonstrate that the KL basis approximates the optimal basis within a factor of $1 + O(1/\sqrt{d})$, with its reconstruction performance asymptotically approaching optimality as the ambient dimension grows—thus addressing the limitations of existing constant-factor error bounds in terms of algorithmic applicability.

Abstract

Principal component analysis (PCA) is optimal for the linear reconstruction of Gaussian data, a foundational property underlying its central role in algorithms and signal processing. Its nonlinear analogue, however, is notoriously subtle: in 2011, Mallat and Zeitouni conjectured that the Karhunen--Loève (KL) basis remains optimal even when the retained coordinates are chosen adaptively per sample, a property that would theoretically justify the ubiquitous pipeline of PCA followed by sparse thresholding. In this paper, we establish a $1+O(1/\sqrt{d})$-approximate version of the retained-energy form of the Mallat--Zeitouni conjecture, showing that the KL basis is within this factor of the optimal basis. This dimension-free comparison depends only on the number of retained coordinates and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows. It complements the universal-constant reconstruction-error comparison of Litvak and Tikhomirov (Ann. Appl. Probab., 2018), while providing a comparison naturally suited for algorithmic analysis. Our proof rests on a clean, conceptual reduction: we relax arbitrary rotations to a deterministic threshold bound via Schur--Horn majorization, and identify the remaining loss with the correlation gap of the rank-$d$ uniform matroid over Gaussian level sets.

Problem

• nonlinear PCA
• Gaussian data
• Karhunen–Loève basis
• Mallat–Zeitouni conjecture
• correlation gap

Innovation

• nonlinear PCA
• Karhunen–Loève basis
• correlation gap
• Schur–Horn majorization
• Gaussian data