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