Sparse principal component analysis via axis-aligned random projections

TL;DR

This paper presents a non-iterative method for sparse PCA using axis-aligned random projections, achieving minimax optimal rates with theoretical guarantees and strong empirical performance.

Contribution

Introduces a novel non-iterative sparse PCA algorithm based on axis-aligned random projections with proven minimax optimal convergence rates.

Findings

Algorithm attains minimax optimal rate of convergence.

Theoretical analysis reveals the statistical and computational trade-offs.

Numerical studies confirm competitive finite-sample performance.

Abstract

We introduce a new method for sparse principal component analysis, based on the aggregation of eigenvector information from carefully-selected axis-aligned random projections of the sample covariance matrix. Unlike most alternative approaches, our algorithm is non-iterative, so is not vulnerable to a bad choice of initialisation. We provide theoretical guarantees under which our principal subspace estimator can attain the minimax optimal rate of convergence in polynomial time. In addition, our theory provides a more refined understanding of the statistical and computational trade-off in the problem of sparse principal component estimation, revealing a subtle interplay between the effective sample size and the number of random projections that are required to achieve the minimax optimal rate. Numerical studies provide further insight into the procedure and confirm its highly competitive finite-sample performance.