Polar decomposition based algorithms on the product of Stiefel manifolds with applications in tensor approximation
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In this paper, based on the matrix polar decomposition, we propose a general algorithmic framework to solve a class of optimization problems on the product of Stiefel manifolds. We establish the weak convergence and global convergence of this general algorithmic approach based on the Łojasiewicz gradient inequality. This general algorithm and its convergence results are applied to the best rank-1 approximation, low rank orthogonal approximation and low multilinear rank approximation for higher order tensors. We also present a symmetric variant of this general algorithm to solve a symmetric variant of this class of optimization models, which essentially optimizes over a single Stiefel manifold. We establish its weak convergence and global convergence in a similar way. This symmetric variant and its convergence results are applied to the best symmetric rank-1 approximation and low rank symmetric orthogonal approximation for higher order tensors. It turns out that well-known algorithms such as HOPM, S-HOPM, LROAT, S-LROAT are all special cases of this general algorithmic framework and its symmetric variant.
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