Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descent

TL;DR

This paper introduces a scalable, provably convergent algorithm for tensor robust PCA that efficiently recovers low-rank tensors from corrupted observations using scaled gradient descent and adaptive thresholding.

Contribution

It proposes a novel tensor RPCA method with theoretical guarantees and practical scalability, improving over existing algorithms in efficiency and robustness.

Findings

Linear convergence rate independent of condition number

Outperforms state-of-the-art tensor RPCA algorithms

Effective on synthetic and real-world data

Abstract

An increasing number of data science and machine learning problems rely on computation with tensors, which better capture the multi-way relationships and interactions of data than matrices. When tapping into this critical advantage, a key challenge is to develop computationally efficient and provably correct algorithms for extracting useful information from tensor data that are simultaneously robust to corruptions and ill-conditioning. This paper tackles tensor robust principal component analysis (RPCA), which aims to recover a low-rank tensor from its observations contaminated by sparse corruptions, under the Tucker decomposition. To minimize the computation and memory footprints, we propose to directly recover the low-dimensional tensor factors -- starting from a tailored spectral initialization -- via scaled gradient descent (ScaledGD), coupled with an iteration-varying thresholding operation to adaptively remove the impact of corruptions. Theoretically, we establish that the proposed algorithm converges linearly to the true low-rank tensor at a constant rate that is independent with its condition number, as long as the level of corruptions is not too large. Empirically, we demonstrate that the proposed algorithm achieves better and more scalable performance than state-of-the-art matrix and tensor RPCA algorithms through synthetic experiments and real-world applications.