Tensor Robust Principal Component Analysis: Better recovery with atomic norm regularization

TL;DR

This paper introduces an atomic-norm regularization approach for tensor RPCA, providing improved recovery guarantees and demonstrating superior performance over existing methods, especially for high-rank tensors.

Contribution

It offers new theoretical guarantees for tensor RPCA using atomic-norm regularization and analyzes a nonconvex formulation with globally optimal local minima.

Findings

Atomic-norm regularization improves tensor recovery accuracy.

The nonconvex model reliably recovers high-rank tensors.

Numerical experiments outperform existing algorithms.

Abstract

This paper studies tensor-based Robust Principal Component Analysis (RPCA) using atomic-norm regularization. Given the superposition of a sparse and a low-rank tensor, we present conditions under which it is possible to exactly recover the sparse and low-rank components. Our results improve on existing performance guarantees for tensor-RPCA, including those for matrix RPCA. Our guarantees also show that atomic-norm regularization provides better recovery for tensor-structured data sets than other approaches based on matricization. In addition to these performance guarantees, we study a nonconvex formulation of the tensor atomic-norm and identify a class of local minima of this nonconvex program that are globally optimal. We demonstrate the strong performance of our approach in numerical experiments, where we show that our nonconvex model reliably recovers tensors with ranks larger than all of their side lengths, significantly outperforming other algorithms that require matricization.