A nonlinear high-order transformations-based method for high-order tensor completion

Abstract

The high-order tensor nuclear norm model (HTNN) has recently shown promising results in tensor completion problems. The HTNN-based approaches rely on the low-rank structure of tensor slices during reversible transformations. However, under reversible transformations, the low-rank configuration within the slice-wise modality of the tensor’s structure is not markedly pronounced. In order to more effectively describe the low-rank characteristics, we propose a model based on the nonlinear high-order transform-based tensor nuclear norm (NHTNN). Specifically, our framework consists of a linearly semi-orthogonal transformation along the high-dimensional modality and an element-wise nonlinear transformation. We introduce a model for tensor completion, grounded in the suggested measure of tensor low-rank, i.e., NHTNN. Utilizing this non-convex nonlinear model, we formulate a proximal alternating minimization (PAM) algorithm, establishing its convergence through a rigorous proof. In experiments on datasets such as hyperspectral videos (HSVs) and color videos (CVs), our approach demonstrates superior quantitative numerical results and qualitative visual effects compared to cutting-edge tensor completion techniques.