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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 corrections to the subproblems rather than recomputing the tensor contractions. This approximation is accurate when the factor matrices are changing little across iterations, which occurs when alternating least squares approaches convergence. We provide a theoretical analysis to bound the approximation error. Our numerical experiments demonstrate that the proposed pairwise perturbation algorithms are easy to control and converge to minima that are as good as alternating least squares. The experimental results show improvements of up to 3.1X with respect to state-of-the-art alternating least squares approaches for various model tensor problems and real datasets.
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
Alternating least squares is the most widely used algorithm for CP tensor decomposition. However, alternating least squares may exhibit slow or no convergence, especially when high accuracy is required. An alternative approach is to regard CP decompo
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 s
We consider best approximation problems in a nonlinear subset $mathcal{M}$ of a Banach space of functions $(mathcal{V},|bullet|)$. The norm is assumed to be a generalization of the $L^2$-norm for which only a weighted Monte Carlo estimate $|bullet|_n
There are plenty of applications and analysis for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting n