Sparse Principal Components Analysis

TL;DR

This paper introduces a sparse PCA algorithm that selects high-variance coordinates, estimates eigenvectors, and thresholds them, demonstrating consistency under sparsity assumptions in a single factor model, with applications to ECG data.

Contribution

The paper proposes a simple sparse PCA method with theoretical consistency guarantees under sparsity assumptions, extending classical PCA analysis.

Findings

The sparse PCA algorithm effectively identifies principal factors in sparse signal representations.

Theoretical proof of consistency for the sparse PCA estimator in a single factor model.

Application to ECG data illustrates practical utility of the method.

Abstract

Principal components analysis (PCA) is a classical method for the reduction of dimensionality of data in the form of n observations (or cases) of a vector with p variables. For a simple model of factor analysis type, it is proved that ordinary PCA can produce a consistent (for n large) estimate of the principal factor if and only if p(n) is asymptotically of smaller order than n. There may be a basis in which typical signals have sparse representations: most co-ordinates have small signal energies. If such a basis (e.g. wavelets) is used to represent the signals, then the variation in many coordinates is likely to be small. Consequently, we study a simple "sparse PCA" algorithm: select a subset of coordinates of largest variance, estimate eigenvectors from PCA on the selected subset, threshold and reexpress in the original basis. We illustrate the algorithm on some exercise ECG data, and prove that in a single factor model, under an appropriate sparsity assumption, it yields consistent estimates of the principal factor.