Robust PCA and subspace tracking from incomplete observations using L0-surrogates

TL;DR

This paper introduces a novel Grassmannian-based algorithm for robustly decomposing and tracking low-rank matrices from incomplete, corrupted data without prior outlier information, outperforming existing convex relaxation methods.

Contribution

It presents a non-convex, Grassmannian conjugate gradient approach that improves robustness and handles higher outlier levels and matrix ranks compared to state-of-the-art methods.

Findings

Handles more outliers than existing methods

Tracks subspaces with higher rank from incomplete data

Uses non-convex sparsity measures for better outlier detection

Abstract

Many applications in data analysis rely on the decomposition of a data matrix into a low-rank and a sparse component. Existing methods that tackle this task use the nuclear norm and L1-cost functions as convex relaxations of the rank constraint and the sparsity measure, respectively, or employ thresholding techniques. We propose a method that allows for reconstructing and tracking a subspace of upper-bounded dimension from incomplete and corrupted observations. It does not require any a priori information about the number of outliers. The core of our algorithm is an intrinsic Conjugate Gradient method on the set of orthogonal projection matrices, the so-called Grassmannian. Non-convex sparsity measures are used for outlier detection, which leads to improved performance in terms of robustly recovering and tracking the low-rank matrix. In particular, our approach can cope with more outliers and with an underlying matrix of higher rank than other state-of-the-art methods.