Bing Zheng

Maolin Liang, Bing Zheng

In this paper, an iterative algorithm is presented for solving Sylvester tensor equation $\mathscr{A}*_M\mathscr{X}+\mathscr{X}*_N\mathscr{C}=\mathscr{D}$, where $\mathscr{A}$, $\mathscr{C}$ and $\mathscr{D}$ are given tensors with appropriate sizes, and the symbol $*_N$ denotes the Einstein product. By this algorithm, the solvability of this tensor equation can be determined automatically, and the solution of which (when it is solvable) can be derived within finite iteration steps for any initial iteration tensors in the absence of roundoff errors. Particularly, the least F-norm solution of the aforementioned equation can be derived by choosing special initial iteration tensors. As application, we apply the proposed algorithm to the tensor nearness problem related to the Sylvester tensor equation mentioned above. It is proved that the solution to this problem can also be obtained within finite iteration steps by solving another Sylvester tensor equation. The performed numerical experiments show that the algorithm we propose here is promising.

Hongbing Zhang, Bing Zheng

The low-rank tensor completion (LRTC) problem aims to reconstruct a tensor from partial sample information, which has attracted significant interest in a wide range of practical applications such as image processing and computer vision. Among the various techniques employed for the LRTC problem, non-convex relaxation methods have been widely studied for their effectiveness in handling tensor singular values, which are crucial for accurate tensor recovery. While the minimax concave penalty (MCP) non-convex relaxation method has achieved promising results in tackling the LRTC problem and gained widely adopted, it exhibits a notable limitation: insufficient penalty on small singular values during the singular value handling process, resulting in inefficient tensor recovery. To address this issue and enhance recovery performance, a novel minimax $p$-th order concave penalty (MPCP) function is proposed. Based on this novel function, a tensor $p$-th order $τ$ norm is proposed as a non-convex relaxation for tensor rank approximation, thereby establishing an MPCP-based LRTC model. Furthermore, theoretical convergence guarantees are rigorously established for the proposed method. Extensive numerical experiments conducted on multiple real datasets demonstrate that the proposed method outperforms the state-of-the-art methods in both visual quality and quantitative metrics.