Optimal low rank tensor recovery

TL;DR

This paper establishes theoretical guarantees and efficient algorithms for exact recovery of high-order low-rank tensors from minimal entries, with applications demonstrated on hyperspectral image data.

Contribution

It provides the first rigorous sample complexity bounds for tensor recovery using Riemannian optimization and spectral initialization.

Findings

High probability exact recovery from as few as O((r^d+dnr)log(d)) entries.

Order 3 tensors can be recovered with as few as O(nr) entries.

Numerical experiments confirm the effectiveness of the proposed methods.

Abstract

We investigate the sample size requirement for exact recovery of a high order tensor of low rank from a subset of its entries. In the Tucker decomposition framework, we show that the Riemannian optimization algorithm with initial value obtained from a spectral method can reconstruct a tensor of size $n\times n \times\cdots \times n$ tensor of ranks $(r,\cdots,r)$ with high probability from as few as $O((r^d+dnr)\log(d))$ entries. In the case of order 3 tensor, the entries can be asymptotically as few as $O(nr)$ for a low rank large tensor. We show the theoretical guarantee condition for the recovery. The analysis relies on the tensor restricted isometry property (tensor RIP) and the curvature of the low rank tensor manifold. Our algorithm is computationally efficient and easy to implement. Numerical results verify that the algorithms are able to recover a low rank tensor from minimum number of measurements. The experiments on hyperspectral images recovery also show that our algorithm is capable of real world signal processing problems.