Provable Near-Optimal Low-Multilinear-Rank Tensor Recovery

TL;DR

This paper introduces a provably efficient algorithm for recovering low-multilinear-rank tensors from minimal measurements, achieving near-optimal sampling complexity and high computational efficiency.

Contribution

It presents a Riemannian gradient algorithm with optimal measurement bounds for tensor recovery, leveraging tensor restricted isometry property and manifold geometry.

Findings

Reconstructs tensors with high probability from O(nr^2 + r^{d+1}) measurements.

Achieves optimal sampling complexity in the dimension n.

Uses efficient higher order SVD on small tensors for computation.

Abstract

We consider the problem of recovering a low-multilinear-rank tensor from a small amount of linear measurements. We show that the Riemannian gradient algorithm initialized by one step of iterative hard thresholding can reconstruct an order-$d$ tensor of size $n\times\ldots\times n$ and multilinear rank $(r,\ldots,r)$ with high probability from only $O(nr^2 + r^{d+1})$ measurements, assuming $d$ is a constant. This sampling complexity is optimal in $n$, compared to existing results whose sampling complexities are all unnecessarily large in $n$. The analysis relies on the tensor restricted isometry property (TRIP) and the geometry of the manifold of all tensors with a fixed multilinear rank. High computational efficiency of our algorithm is also achieved by doing higher order singular value decomposition on intermediate small tensors of size only $2r\times \ldots\times 2r$ rather than on tensors of size $n\times \ldots\times n$ as usual.