Paper:
Robust Tensor Decomposition with Gross Corruption
In this paper, we study the statistical performance of robust tensor decomposition with gross corruption. The observations are noisy realization of the superposition of a low-rank tensor W * and an entrywise sparse corruption tensor V * . Unlike conventional noise with bounded variance in previous convex tensor decomposition analysis, the magnitude of the gross corruption can be arbitrary large. We show that under certain conditions, the true low-rank tensor as well as the sparse corruption tensor can be recovered simultaneously. Our theory yields nonasymptotic Frobenius-norm estimation error bounds for each tensor separately. We show through numerical experiments that our theory can precisely predict the scaling behavior in practice.

Review:
This paper studies low rank tensor recovery problem with gross corruption using convex program. The convex programs considered here seem to be inspired by matrix completion and tensor trace norms. The main contribution is in handling gross corruption, which is a very natural noise model. The paper also involves some synthetic experiments.Overall this is an interesting paper. The analysis is clean. The results are reasonable but the bounds involves many parameters and would be good if the authors can give more intuitions on different trade-offs. For example, although Assumption 1 is reasonable (e.g. it holds for random subspace \Omega and random sparsity pattern S as long as the rank/sparsity is not too large), it is not immediately clear and would be good to discuss some typical cases where it is true. It would also be good if the bound can be simplified (for example in the case where all the n_k's are the same, r_k's are also the same) to provide more intuition.Another (slight) concern is that the authors are mixing tensor decomposition with tensor completion in the intro. Although these are closely related, tensor decomposition (especially CP decomposition) is often harder than tensor completion (also for low rank tensor completion typically the definition of rank is different from CP rank, as in this paper).