Sparse PCA: Phase Transitions in the Critical Regime

TL;DR

This paper analyzes phase transitions in sparse PCA within high-dimensional settings, revealing thresholds for signal detectability and support recovery using kernel PCA, and identifying optimal kernel functions for these tasks.

Contribution

It introduces a detailed phase transition analysis for sparse PCA in the critical regime, extending kernel PCA methods and identifying optimal kernels for detection and support recovery.

Findings

Detectability phase transition analogous to BBP transition

Support recovery is possible above the detection threshold

Soft thresholding kernel is nearly optimal for detection

Abstract

This work studies estimation of sparse principal components in high dimensions. Specifically, we consider a class of estimators based on kernel PCA, generalizing the covariance thresholding algorithm proposed by Krauthgamer et al. (2015). Focusing on Johnstone's spiked covariance model, we investigate the "critical" sparsity regime, where the sparsity level $m$, sample size $n$, and dimension $p$ each diverge and $m/\sqrt{n} \rightarrow \beta$, $p/n \rightarrow \gamma$. Within this framework, we develop a fine-grained understanding of signal detection and recovery. Our results establish a detectability phase transition, analogous to the Baik--Ben Arous--P\'ech\'e (BBP) transition: above a certain threshold -- depending on the kernel function, $\gamma$, and $\beta$ -- kernel PCA is informative. Conversely, below the threshold, kernel principal components are asymptotically orthogonal to the signal. Notably, above this detection threshold, we find that consistent support recovery is possible with high probability. Sparsity plays a key role in our analysis, and results in more nuanced phenomena than in related studies of kernel PCA with delocalized (dense) components. Finally, we identify optimal kernel functions for detection -- and consequently, support recovery -- and numerical calculations suggest that soft thresholding is nearly optimal.