Rank-1 Tensor Properties with Applications to a Class of Tensor Optimization Problems
研究了基于张量与特定展开矩阵间秩一等价性质的一类张量优化问题的模型与算法，提出了三种求解最佳秩一张量逼近问题的松弛方法，包括两种凸松弛和一种非凸松弛。
© 2016 Society for Industrial and Applied Mathematics. This paper studies models and algorithms for a class of tensor optimization problems, based on a rank-1 equivalence property between a tensor and certain unfoldings. It is first shown that in dth order tensor space, the set of rank-1 tensors is the same as the intersection of ⌈log2(d)⌉ tensor sets, of which tensors have a specific rank-1 balanced unfolding matrix. Moreover, the number ⌈log2(d)⌉ is proved to be optimal in some sense. Based on the above equivalence property, three relaxation approaches for solving the best rank-1 tensor approximation problems are proposed, including two convex relaxations and a nonconvex one. The two convex relaxations utilize the matrix nuclear norm regularization/constraints. They have the advantage of identifying whether the solution is a global optimizer of the original problem, by computing the nuclear norm or the Frobenius norm of a certain matrix. Under certain assumptions, the optimal solution of the original problem is characterized by the solution to the dual of the nuclear norm constrained problem. The nonconvex relaxation can be solved via the conventional alternating minimization scheme, with the output being always a rank-1 tensor. Numerical experiments demonstrate the effectiveness of the proposed methods.