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arXiv · 1709.00033

A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem

Paul Breiding·Nick Vannieuwenhoven
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Abstract

The canonical tensor rank approximation problem (TAP) consists of approximating a real-valued tensor by one of low canonical rank, which is a challenging non-linear, non-convex, constrained optimization problem, where the constraint set forms a non-smooth semi-algebraic set. We introduce a Riemannian Gauss-Newton method with trust region for solving small-scale, dense TAPs. The novelty of our approach is threefold. First, we parametrize the constraint set as the Cartesian product of Segre manifolds, hereby formulating the TAP as a Riemannian optimization problem, and we argue why this parametrization is among the theoretically best possible. Second, an original ST-HOSVD-based retraction operator is proposed. Third, we introduce a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and which often yields a quick escape path from such spurious decompositions. Numerical experiments show improvements of up to three orders of magnitude in terms of the expected time to compute a successful solution over existing state-of-the-art methods.

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Paul Breiding, Nick Vannieuwenhoven. 2018-07-11. A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem. https://doi.org/10.1137/17m114618x

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