A Hybrid Norm for Guaranteed Tensor Recovery

Abstract

Benefiting from the superiority of tensor Singular Value Decomposition (t-SVD) in excavating low-rankness in the spectral domain over other tensor decompositions (like Tucker decomposition), t-SVD-based tensor learning has shown promising performance and become an emerging research topic in computer vision and machine learning very recently. However, focusing on modeling spectral low-rankness, the t-SVD-based models may be insufficient to exploit low-rankness in the original domain, leading to limited performance while learning from tensor data (like videos) that are low-rank in both original and spectral domains. To this point, we define a hybrid tensor norm dubbed the “Tubal + Tucker” Nuclear Norm (T2NN) as the sum of two tensor norms, respectively, induced by t-SVD and Tucker decomposition to simultaneously impose low-rankness in both spectral and original domains. We further utilize the new norm for tensor recovery from linear observations by formulating a penalized least squares estimator. The statistical performance of the proposed estimator is then analyzed by establishing upper bounds on the estimation error in both deterministic and non-asymptotic manners. We also develop an efficient algorithm within the framework of Alternating Direction Method of Multipliers (ADMM). Experimental results on both synthetic and real datasets show the effectiveness of the proposed model.