Robust Tensor Decomposition via Orientation Invariant Tubal Nuclear Norms

Aiming at recovering an unknown tensor (i.e., multi-way array) corrupted by both sparse outliers and dense noises, robust tensor decomposition (RTD) serves as a powerful pre-processing tool for subsequent tasks like classification and target detection in many computer vision and machine learning applications. Recently, tubal nuclear norm (TNN) based optimization is proposed with superior performance as compared with other tensorial nuclear norms for tensor recovery. However, one major limitation is its orientation sensitivity due to low-rankness strictly defined along tubal orientation and it cannot simultaneously model spectral low-rankness in multiple orientations. To this end, we introduce two new tensor norms called OITNN-O and OITNN-L to exploit multi-orientational spectral low-rankness for an arbitrary K-way (K ≥ 3) tensors. We further formulate two RTD models via the proposed norms and develop two algorithms as the solutions. Theoretically, we establish non-asymptotic error bounds which can predict the scaling behavior of the estimation error. Experiments on real-world datasets demonstrate the superiority and effectiveness of the proposed norms.