ACCELERATING ILL-CONDITIONED ROBUST LOW-RANK TENSOR REGRESSION

An important problem that arises across different applications in signal processing, machine learning, and data science is to reliably estimate a tensor from a small number of measurements that are possibly corrupted. Leveraging the low-rank structure under the Tucker decomposition, we propose a provably efficient algorithm that directly estimates the tensor factors by solving a nonsmooth and non-convex composite optimization problem that minimizes the least absolute deviation loss. The proposed algorithm-built on subgradient methods-harnesses preconditioners that are designed to be equivariant w.r.t. the low-rank parameterization, and is shown to achieve local linear convergence at a constant rate under the Gaussian design. Numerical experiments are provided to corroborate the superior performance of the proposed algorithm.