461 RWTH Publication No: 711138        2017        IGPM461.pdf
TITLE Stable ALS Approximation in the TT-Format for Rank-Adaptive Tensor Completion
AUTHORS Lars Grasedyck, Sebastian Krämer
ABSTRACT Low rank tensor completion is a highly ill-posed inverse problem, particularly when the data model is not accurate, and some sort of regularization is required in order to solve it. In this article we focus on the calibration of the data model. For alternating optimization, we observe that existing rank adaption methods do not enable a continuous transition between manifolds of different ranks. We denote this flaw as instability (under truncation). As a consequence of this flaw, arbitrarily small changes in the singular values of an iterate can have arbitrarily large influence on the further reconstruction. We therefore introduce a singular value based reg- ularization to the standard alternating least squares (ALS), which is motivated by averaging in micro-steps. We prove its stability and derive a natural semi-implicit rank adaption strategy. We further prove that the standard ALS micro-steps are only stable on manifolds of fixed ranks, and only around points that have what we define as internal tensor restricted isometry property iTRIP. Finally, we provide numerical examples that show improvements of the reconstruction quality up to orders of magnitude in the new Stable ALS Approximation (SALSA) compared to standard ALS.
KEYWORDS tensor completion, MPS, tensor train, TT, hierarchical Tucker, HT, alternating optimization, ALS, high-dimensional, low rank, SVD, ill-posedness, stability
DOI 10.1007/s00211-019-01072-4
PUBLICATION Numerische Mathematik
143(4), pp 855-904