Fast Robust Subspace Tracking via PCA in Sparse Data-Dependent Noise

TL;DR

This paper presents a fast, robust subspace tracking algorithm capable of handling sparse outliers in data, with provable correctness and near-optimal delay, while maintaining computational efficiency comparable to standard PCA.

Contribution

It introduces a novel fast mini-batch robust subspace tracking method with theoretical guarantees, extending robust PCA to time-varying subspaces with sparse data-dependent noise.

Findings

Algorithm tracks subspace with near-optimal delay

Provides non-asymptotic guarantees for PCA in data-dependent noise

Achieves computational complexity similar to standard PCA

Abstract

This work studies the robust subspace tracking (ST) problem. Robust ST can be simply understood as a (slow) time-varying subspace extension of robust PCA. It assumes that the true data lies in a low-dimensional subspace that is either fixed or changes slowly with time. The goal is to track the changing subspaces over time in the presence of additive sparse outliers and to do this quickly (with a short delay). We introduce a "fast" mini-batch robust ST solution that is provably correct under mild assumptions. Here "fast" means two things: (i) the subspace changes can be detected and the subspaces can be tracked with near-optimal delay, and (ii) the time complexity of doing this is the same as that of simple (non-robust) PCA. Our main result assumes piecewise constant subspaces (needed for identifiability), but we also provide a corollary for the case when there is a little change at each time. A second contribution is a novel non-asymptotic guarantee for PCA in linearly data-dependent noise. An important setting where this is useful is for linearly data dependent noise that is sparse with support that changes enough over time. The analysis of the subspace update step of our proposed robust ST solution uses this result.