Slim Is Better: Transform-Based Tensor Robust Principal Component Analysis

This paper addresses the tensor robust principal component analysis (RPCA) by employing linear slim transforms along the mode-3 of the tensor. Previous works have empirically shown the superiority of slim transforms over traditional square ones in low-rank tensor recovery. However, the recovery guarantee for the slim transform-based tensor RPCA (SRPCA) remains an unresolved issue, as existing guarantees are only applicable to invertible, inner product preserving, and self-adjoint transforms. In contrast, we establish the recovery guarantee for SRPCA that is applicable to any mode-3 linear slim transform under certain conditions. Specifically, new tensor incoherence conditions are deduced to accommodate slim transforms and can also be simplified to the existing conditions pertaining to the discrete Fourier transform. Our theoretical analysis reveals that the slim transform with a condition number of 1 enjoys an averaging effect on tensor incoherence parameters through its composing square transforms, thus leading to a more relaxed recovery bound for SRPCA compared to its square counterparts. This insight is validated through experimental results on both synthetic and real data, which demonstrate the improved performance of SRPCA over traditionally square transform-based tensor RPCA.