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

arXiv: https://arxiv.org/abs/1510.04747 · HTML: https://mlsanity.com/paper/1510.04747/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Oct 2015.

## Built on by (most cited first, 5 of 5)

- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descent](https://mlsanity.com/paper/2206.09109/) (2022)
- [Fast Robust Tensor Principal Component Analysis via Fiber CUR Decomposition](https://mlsanity.com/paper/2108.10448/) (2021)
- [Deep Unfolded Tensor Robust PCA with Self-supervised Learning](https://mlsanity.com/paper/2212.11346/) (2022)
- [Optimal vintage factor analysis with deflation varimax](https://mlsanity.com/paper/2310.10545/) (2023)

## Abstract (arXiv)

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.