Tensor Completion via A Generalized Transformed Tensor T-Product Decomposition Without t-SVD

摘要

Abstract Matrix and tensor nuclear norms have been successfully used to promote the low-rankness of tensors in low-rank tensor completion. However, singular value decomposition (SVD), which is computationally expensive for large-scale matrices, frequently appears in solving those nuclear norm minimization models. Based on the tensor-tensor product (T-product), in this paper, we first establish the equivalence between the so-called transformed tubal nuclear norm for a third-order tensor and the minimum of the sum of two factor tensors’ squared Frobenius norms under a general invertible linear transform. Gainfully, we introduce a mode-unfolding (often named as “spatio-temporal” in the internet traffic data recovery literature) regularized tensor completion model that is able to efficiently exploit the hidden structures of tensors. Then, we propose an implementable alternating minimization algorithm to solve the underlying optimization model. It is remarkable that our approach does not require any SVDs and all subproblems of our algorithm enjoy closed-form solutions. A series of numerical experiments on traffic data recovery, color images and videos inpainting demonstrate that our SVD-free approach takes less computing time to achieve satisfactory accuracy than some state-of-the-art tensor nuclear norm minimization approaches.