Smooth hard shrinkage operator for tensor completion based on self-adaptive transforms

The tensor completion(TC) problem has attracted wide attention due to its significant practical significance and broad application background. Various approaches have been used to solve this problem, including approximating the rank function through its convex envelope, the tensor nuclear norm. However, due to the gap between the rank function and its convex envelope, this approach often fails to achieve satisfactory recovery. Recent developments have highlighted that non-explicit non-convex functions, especially the one induced by the Smooth Hard(SH) shrinkage operator, can approximate the l0 norm more effectively compared to other methods, and this function facilitates the reconstruction of noisy MRI data with minimal sampling rate requirements. In our paper, we implement the SH shrinkage operator within the TC framework, achieving accurate tensor data recovery at lower sampling rates. Moreover, we adopt a method that adaptively generates transformation matrices based on specific features of the data, which, compared to traditional TC models, can better exploit the inherent low-rank properties of tensor data. Additionally, our approach integrates a reweighting strategy to efficiently merge tensors recovered from various dimensions. Finally, a feasible minimization algorithm is proposed, which combines the non-explicit non-convex regularizer and self-adaptive transforms. It is proven that the constructed sequence of the proposed algorithm is a Cauchy sequence and can converge to a stable point. Extensive experimental results demonstrate that our proposed approach can achieve accurate tensor recovery at lower sampling rates compared to other methods.