# Provable Dynamic Robust PCA or Robust Subspace Tracking

## TL;DR

This paper introduces simple-ReProCS, a novel online algorithm for dynamic robust PCA that effectively tracks changing low-dimensional subspaces in data with sparse outliers, under weaker assumptions than previous methods.

## Contribution

It provides the first theoretical guarantee for dynamic RPCA under relaxed assumptions, improving outlier tolerance and demonstrating online, fast, and memory-efficient performance.

## Key findings

- Guarantees dynamic RPCA under weakened assumptions.
- Achieves higher outlier tolerance than previous methods.
- Operates online with near-optimal memory complexity.

## Abstract

Dynamic robust PCA refers to the dynamic (time-varying) extension of robust PCA (RPCA). It assumes that the true (uncorrupted) data lies in a low-dimensional subspace that can change with time, albeit slowly. The goal is to track this changing subspace over time in the presence of sparse outliers. We develop and study a novel algorithm, that we call simple-ReProCS, based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. Our work provides the first guarantee for dynamic RPCA that holds under weakened versions of standard RPCA assumptions, slow subspace change and a lower bound assumption on most outlier magnitudes. Our result is significant because (i) it removes the strong assumptions needed by the two previous complete guarantees for ReProCS-based algorithms; (ii) it shows that it is possible to achieve significantly improved outlier tolerance, compared with all existing RPCA or dynamic RPCA solutions by exploiting the above two simple extra assumptions; and (iii) it proves that simple-ReProCS is online (after initialization), fast, and, has near-optimal memory complexity.