Robust Low-Rank Tensor Train Recovery

Tensor train (TT) decomposition represents an N-order tensor using $O(N)$ 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 an $\ell_{1}$ loss function to enhance robustness against outliers. We first establish the $\ell_{1}/\ell_{2}$-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 with N. We also prove the sharpness property for the $\ell_{1}$ loss function optimized over TT format tensors. Building on the $\ell_{1}/\ell_{2}$-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.