The Rank-1 Completion Problem for Cubic Tensors

2024/04/12 by Jinling Zhou, Zhou, Jinling, Jiawang Nie +4 · 2 citations
Engineering · Mathematics · #Elasticity and Material Modeling #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2404.08171

openalex publication_date 2024/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper studies the rank-1 tensor completion problem for cubic tensors. First of all, we show that this problem is equivalent to a special rank-1 matrix recovery problem. When the tensor is strongly rank-1 completable, we show that the problem is equivalent to a rank-1 matrix completion problem and it can be solved by an iterative formula. For other cases, we propose both nuclear norm relaxation and moment relaxation methods for solving the resulting rank-1 matrix recovery problem. The nuclear norm relaxation sometimes returns a rank-1 tensor completion, while sometimes it does not. When it fails, we apply the moment hierarchy of semidefinite programming relaxations to solve the rank-1 matrix recovery problem. The moment hierarchy can always get a rank-1 tensor completion, or detect its nonexistence. Numerical experiments are shown to demonstrate the efficiency of these proposed methods.

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