Minimum $n$-Rank Approximation via Iterative Hard Thresholding

The problem of recovering a low $n$-rank tensor is an extension of sparse recovery problem from the low dimensional space (matrix space) to the high dimensional space (tensor space) and has many applications in computer vision and graphics such as image inpainting and video inpainting. In this paper, we consider a new tensor recovery model, named as minimum $n$-rank approximation (MnRA), and propose an appropriate iterative hard thresholding algorithm with giving the upper bound of the $n$-rank in advance. The convergence analysis of the proposed algorithm is also presented. Particularly, we show that for the noiseless case, the linear convergence with rate $\frac{1}{2}$ can be obtained for the proposed algorithm under proper conditions. Additionally, combining an effective heuristic for determining $n$-rank, we can also apply the proposed algorithm to solve MnRA when $n$-rank is unknown in advance. Some preliminary numerical results on randomly generated and real low $n$-rank tensor completion problems are reported, which show the efficiency of the proposed algorithms.