Phase Transitions in Sparse PCA

TL;DR

This paper investigates the limits of sparse PCA estimation, revealing phase transitions where optimal estimation is possible but computationally hard, especially in high-rank and low-density regimes.

Contribution

It analyzes the phase transitions in sparse PCA, showing the gap between information-theoretic limits and algorithmic feasibility across different regimes.

Findings

AMP achieves asymptotic optimality for rank one with high density

Existence of a parameter region where estimation is information-theoretically possible but computationally hard

Phase transitions depend on rank and sparsity, affecting algorithm performance

Abstract

We study optimal estimation for sparse principal component analysis when the number of non-zero elements is small but on the same order as the dimension of the data. We employ approximate message passing (AMP) algorithm and its state evolution to analyze what is the information theoretically minimal mean-squared error and the one achieved by AMP in the limit of large sizes. For a special case of rank one and large enough density of non-zeros Deshpande and Montanari [1] proved that AMP is asymptotically optimal. We show that both for low density and for large rank the problem undergoes a series of phase transitions suggesting existence of a region of parameters where estimation is information theoretically possible, but AMP (and presumably every other polynomial algorithm) fails. The analysis of the large rank limit is particularly instructive.