Year: 2025
Type: article
Abstract: Tensor train (TT) decomposition represents anN-order tensor usingO(N) matrices (i.e., factors) of small dimensions, achieved through products among these factors. Due to its compact representation, TT decomposition has found wide applications, including various tensor recovery problems in signal processing and quantum information. In this paper, we study the problem of reconstructing a TT format tensor from measurements that are contaminated by outliers with arbitrary values. Given the vulnerability of smooth formulations to corruptions, we use anl1loss function to enhance robustness against outliers. We first establish thel1/l2-restricted isometry property (RIP) for Gaussian measurement operators, demonstrating that the information in the TT format tensor can be preserved using a number of measurements that grows linearly withN. We also prove the sharpness property for thel1loss function optimized over TT format tensors. Building on thel1/l2-RIP and sharpness property, we then propose two complementary methods to recover the TT format tensor from the corrupted measurements: the projected subgradient method (PSubGM), which optimizes over the entire tensor, and the factorized Riemannian subgradient method (FRSubGM), which optimizes directly over the factors. Compared to PSubGM, the factorized approach FRSubGM significantly reduces the memory cost at the expense of a slightly slower convergence rate. Nevertheless, we show that both methods, with diminishing step sizes, converge linearly to the ground-truth tensor given an appropriate initialization, which can be obtained by a truncated spectral method. To the best of our knowledge, this is the first work to provide a theoretical analysis of the robust TT recovery problem and to demonstrate that TT-format tensors can be robustly recovered even when a certain fraction of measurements are arbitrarily corrupted. We conduct various numerical experiments to demonstrate the effectiveness of the two methods in robust TT recovery.
Authors: Zhen Qin, Zhihui Zhu
Institution: The Ohio State University
Language: English
Subfield: Computational Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4409494004