Online Learning for Sparse PCA in High Dimensions: Exact Dynamics and Phase Transitions

TL;DR

This paper analyzes an online sparse PCA algorithm's exact high-dimensional dynamics, revealing phase transitions and providing precise asymptotic performance metrics through a PDE-based approach.

Contribution

It introduces a PDE characterization of the online sparse PCA algorithm's dynamics, enabling exact asymptotic analysis and phase transition detection.

Findings

Exact asymptotic performance metrics derived from PDE analysis

Identification of phase transition phenomena in sparse support recovery

Numerical validation confirms theoretical predictions for moderate dimensions

Abstract

We study the dynamics of an online algorithm for learning a sparse leading eigenvector from samples generated from a spiked covariance model. This algorithm combines the classical Oja's method for online PCA with an element-wise nonlinearity at each iteration to promote sparsity. In the high-dimensional limit, the joint empirical measure of the underlying sparse eigenvector and its estimate provided by the algorithm is shown to converge weakly to a deterministic, measure-valued process. This scaling limit is characterized as the unique solution of a nonlinear PDE, and it provides exact information regarding the asymptotic performance of the algorithm. For example, performance metrics such as the cosine similarity and the misclassification rate in sparse support recovery can be obtained by examining the limiting dynamics. A steady-state analysis of the nonlinear PDE also reveals an interesting phase transition phenomenon. Although our analysis is asymptotic in nature, numerical simulations show that the theoretical predictions are accurate for moderate signal dimensions.