An optimization approach for dynamical Tucker tensor approximation

Abstract

An optimization-based approach for Tucker tensor approximation of parameter-dependent data tensors and solutions of tensor differential equations with low Tucker rank is presented. The problem of updating the tensor decomposition is reformulated as a fitting problem subject to the tangent space without relying on an orthogonality gauge condition. A discrete Euler scheme is established in an alternating least squares framework, where the quadratic subproblems are reduced to trace optimization problems, that are shown to be explicitly solvable and accessible using SVD of small size. In the presence of small singular values, instability for larger ranks is reduced, since the method does not need the (pseudo) inverse of matricizations of the core tensor. Regularization of Tikhonov type can be used to compensate for the lack of uniqueness in the tangent space. The method is validated numerically and shown to be stable also for larger ranks in the case of small singular values of the core unfoldings. Higher order explicit integrators of Runge-Kutta type can be composed. Keywords: Dynamical tensor approximation, Low-rank approximation, Orthogonality gauge condition, Small singular values, Tensor differential equations, Model order reduction