Stable ALS Approximation in the TT-Format for Rank-Adaptive Tensor Completion
For practitioners of tensor completion, this work provides a stable and effective rank-adaptive method, addressing a known instability issue in alternating optimization.
The paper addresses instability in rank-adaptive tensor completion using alternating least squares (ALS) in the TT-format, introducing a singular value-based regularization (SALSA) that ensures stability and enables semi-implicit rank adaption. Numerical experiments show reconstruction quality improvements up to orders of magnitude over standard ALS and RTTC.
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 characteristic as $\textit{instability (under truncation)}$. As a consequence of this property, arbitrarily small changes in the iterate can have arbitrarily large influence on the further reconstruction. We therefore introduce a singular value based regularization to the standard alternating least squares (ALS), which is motivated by averaging in microsteps. We prove its $\textit{stability}$ and derive a natural semi-implicit rank adaption strategy. We further prove that the standard ALS microsteps for completion problems are only stable on manifolds of fixed ranks, and only around points that have what we define as $\textit{internal tensor restricted isometry property, iTRIP}$. In conclusion, numerical experiments are provided that show improvements of the reconstruction quality up to orders of magnitude in the new Stable ALS Approximation (SALSA) compared to standard ALS and the well known Riemannian optimization RTTC.