Tensor vs Matrix Methods: Robust Tensor Decomposition under Block Sparse Perturbations

TL;DR

This paper introduces a novel non-convex iterative algorithm for robust tensor CP decomposition that guarantees recovery under certain conditions and outperforms matrix-based methods in handling block-structured sparse perturbations.

Contribution

The paper proposes a new non-convex algorithm for robust tensor decomposition with proven convergence and superior robustness to block-structured sparse noise compared to matrix approaches.

Findings

The proposed method guarantees global convergence under natural conditions.

It can handle higher levels of perturbation than matrix methods.

Experimental results validate the robustness and effectiveness of the tensor approach.

Abstract

Robust tensor CP decomposition involves decomposing a tensor into low rank and sparse components. We propose a novel non-convex iterative algorithm with guaranteed recovery. It alternates between low-rank CP decomposition through gradient ascent (a variant of the tensor power method), and hard thresholding of the residual. We prove convergence to the globally optimal solution under natural incoherence conditions on the low rank component, and bounded level of sparse perturbations. We compare our method with natural baselines which apply robust matrix PCA either to the {\em flattened} tensor, or to the matrix slices of the tensor. Our method can provably handle a far greater level of perturbation when the sparse tensor is block-structured. This naturally occurs in many applications such as the activity detection task in videos. Our experiments validate these findings. Thus, we establish that tensor methods can tolerate a higher level of gross corruptions compared to matrix methods.