Nonconvex optimization for third‐order tensor completion under wavelet transform

The main aim of this paper is to develop a nonconvex optimization model for third‐order tensor completion under wavelet transform. On the one hand, through wavelet transform of frontal slices, we divide a large tensor data into a main part tensor and three detail part tensors, and the elements of these four tensors are about a quarter of the original tensors. Solving these four small tensors can not only improve the operation efficiency, but also better restore the original tensor data. On the other hand, by using concave correction term, we are able to correct for low rank of tubal nuclear norm (TNN) data fidelity term and sparsity of l1$$ {l}_1 $$ ‐norm data fidelity term. We prove that the proposed algorithm can converge to some critical point. Experimental results on image, magnetic resonance imaging and video inpainting tasks clearly demonstrate the superior performance and efficiency of our developed method over state‐of‐the‐arts including the TNN and other methods.