Rank-1 Tensor Properties with Applications to a Class of Tensor Optimization Problems
Author(s) -
Yuning Yang,
Yunlong Feng,
Xiaolin Huang,
Johan A. K. Suykens
Publication year - 2016
Publication title -
siam journal on optimization
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.066
H-Index - 136
eISSN - 1095-7189
pISSN - 1052-6234
DOI - 10.1137/140983689
Subject(s) - mathematics , matrix norm , tensor (intrinsic definition) , rank (graph theory) , equivalence (formal languages) , norm (philosophy) , regular polygon , relaxation (psychology) , matrix (chemical analysis) , combinatorics , mathematical optimization , pure mathematics , geometry , psychology , social psychology , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , political science , law , composite material
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 $d$th order tensor space, the set of rank-1 tensors is the same as the intersection of $\lceil \log_2(d) \rceil$ tensor sets, of which tensors have a specific rank-1 balanced unfolding matrix. Moreover, the number $\lceil \log_2(d) \rceil$ 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 characterize...
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