Randomized Robust Subspace Recovery for High Dimensional Data Matrices

TL;DR

This paper introduces two randomized sketching methods for robust PCA that significantly reduce computational and memory costs while accurately recovering subspaces in high-dimensional data.

Contribution

It proposes novel randomized designs for robust PCA using data sketching, with theoretical analysis of complexity and recovery guarantees under different outlier models.

Findings

Both methods achieve near-independent complexity from data size.

The approaches provably recover the correct subspace under sparse and independent outlier models.

Numerical simulations confirm the theoretical results on synthetic and real datasets.

Abstract

This paper explores and analyzes two randomized designs for robust Principal Component Analysis (PCA) employing low-dimensional data sketching. In one design, a data sketch is constructed using random column sampling followed by low dimensional embedding, while in the other, sketching is based on random column and row sampling. Both designs are shown to bring about substantial savings in complexity and memory requirements for robust subspace learning over conventional approaches that use the full scale data. A characterization of the sample and computational complexity of both designs is derived in the context of two distinct outlier models, namely, sparse and independent outlier models. The proposed randomized approach can provably recover the correct subspace with computational and sample complexity that are almost independent of the size of the data. The results of the mathematical analysis are confirmed through numerical simulations using both synthetic and real data.