Rank One Completion for Higher Order Tensors

TL;DR

This paper introduces a recursive algorithm for rank one tensor completion that is efficient, accurate, and works for tensors of any order, with proven robustness to small noise.

Contribution

It proposes a novel recursive method for rank one tensor completion, including properties, algorithms, and noise robustness analysis.

Findings

The algorithm efficiently computes rank one tensor completions.

It guarantees uniqueness under certain conditions without noise.

Numerical experiments confirm high accuracy and efficiency.

Abstract

We study the rank one completion problem for tensors of arbitrary orders. The notion of rank one determinable tensors is introduced. We explore its properties and propose a recursive algorithm for computing rank one tensor completion. This algorithm only requires solving linear systems and computing singular vectors. In the absence of noise, it produces a unique rank one completion under some assumptions. In the presence of noise, we show that the computed rank one tensor completion is close to the exact one when the noise is sufficiently small. Numerical experiments demonstrate the efficiency and accuracy of the proposed method.