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

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