Fast and Nonconvex Low Rank Tensor Estimation

This paper investigates the problem of low rank tensor estimation, which aims to recover the true underlying tensor signal from generalized tensor measurements. This problem encompasses a wide range of tasks, such as tensor factorization, tensor completion, and tensor sensing. While existing convex methods suffer from high computational costs, non-convex methods often lack theoretical guarantees. In this work, we are the first to extend the ScaledGD algorithm to the general low rank tensor estimation setting. By analyzing the geometric properties of the loss function $g$, we provide a theoretical analysis of ScaledGD and prove that, under certain assumptions on $g$, it achieves a linear convergence rate. Both synthetic and real-world experiments demonstrate that ScaledGD achieves linear convergence with the lowest computational cost.