ИСТИНА

We propose several algorithms to estimate missing values in visual data based on tensor decompositions. The problem of tensor recovery by a part of its entries has two principal cases. If a tensor is given in all positions but some of the elements were corrupted, the problem is called tensor pursuit. In case of the given mask of corruption it is a tensor completion problem. To make a recovery possible, the initial tensor should have low ranks at least approximately. We deal with tensor ranks defined by ranks of unfold- ing matrices such as Tucker ranks, TT-ranks and others (but not canonical tensor rank). Since in both cases the tensor rank minimization problem is NP-hard, the minimum rank solutions are found by solving a convex opti- mization problem, namely the minimization of the nuclear norms of sets of unfolding matrices. Different sets of unfoldings correspond to different tensor decompositions. For the tensor pursuit problem we also control sparsity via l1 -norm. After that we solve the convex optimization problem. In image recovery we usually have 2 or 3 dimensional data, but it is useful to consider given data as a multidimensional tensor. The crucial parameter here is the way we split indices. We compare several splitting techniques and matrix and tensor recovery methods for MRI data.