Tensor vs Matrix Methods: Robust Tensor Decomposition under Block Sparse Perturbations
This work addresses robust tensor decomposition for applications like video activity detection, offering a significant improvement over matrix methods but is incremental in advancing tensor techniques.
The paper tackles robust CP tensor decomposition in the presence of block sparse perturbations, proposing a non-convex iterative algorithm that provably recovers the low-rank component under incoherence conditions and outperforms matrix-based methods by tolerating higher levels of corruption, with experiments validating these findings.
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.