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The epsilon alternating least squares ($epsilon$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices are assumed to be columnwisely orthonormal. It is shown that the algorithm globally converges to a KKT point for all tensors without any assumption. For the original ALS, by further studying the properties of the polar decomposition, we also establish its global convergence under a reality assumption not stronger than those in the literature. These results completely address a question concerning the global convergence raised in [L. Wang, M. T. Chu and B. Yu, emph{SIAM J. Matrix Anal. Appl.}, 36 (2015), pp. 1--19]. In addition, an initialization procedure is proposed, which possesses a provable lower bound when the number of columnwisely orthonormal factors is one. Armed with this initialization procedure, numerical experiments show that the $epsilon$-ALS exhibits a promising performance in terms of efficiency and effectiveness.
The goal of this work is to fill a gap in [Yang, SIAM J. Matrix Anal. Appl, 41 (2020), 1797--1825]. In that work, an approximation procedure was proposed for orthogonal low-rank tensor approximation; however, the approximation lower bound was only es
In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the existence of
The null distributed controllability of the semilinear heat equation $y_t-Delta y + g(y)=f ,1_{omega}$, assuming that $g$ satisfies the growth condition $g(s)/(vert svert log^{3/2}(1+vert svert))rightarrow 0$ as $vert svert rightarrow infty$ and that
The alternating least squares algorithm for CP and Tucker decomposition is dominated in cost by the tensor contractions necessary to set up the quadratic optimization subproblems. We introduce a novel family of algorithms that uses perturbative corre
We introduce and analyze a space-time least-squares method associated to the unsteady Navier-Stokes system. Weak solution in the two dimensional case and regular solution in the three dimensional case are considered. From any initial guess, we constr