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This work studies the Tensor Robust Principal Component Analysis (TRPCA)problem, which aims to exactly recover the low-rank and sparse components fromtheir sum. Our model is motivated by the recently proposed linear transformsbased tensor-tensor product and tensor SVD. We define a new transforms dependedtensor rank and the corresponding tensor nuclear norm. Then we solve the TRPCAproblem by convex optimization whose objective is a weighted combination of thenew tensor nuclear norm and the $\ell_1$-norm. In theory, we show that undercertain incoherence conditions, the convex program exactly recovers theunderlying low-rank and sparse components. It is of great interest that our newTRPCA model generalizes existing works. In particular, if the studied tensorreduces to a matrix, our TRPCA model reduces to the known matrix RPCA. Our newTRPCA which is allowed to use general linear transforms can be regarded as anextension of our former TRPCA work which uses the discrete Fourier transform.But their proof of the recovery guarantee is different. Numerical experimentsverify our results and the application on image recovery demonstrates thesuperiority of our method.

Exact Recovery of Tensor Robust Principal Component Analysis under Linear Transforms