# The Rank-1 Completion Problem for Cubic Tensors

arXiv: https://arxiv.org/abs/2404.08171 · HTML: https://mlsanity.com/paper/2404.08171/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Apr 2024.

## Built on by (most cited first, 1 of 1)

- [Simple and Nearly-Optimal Sampling for Rank-1 Tensor Completion via Gauss-Jordan](https://mlsanity.com/paper/2408.05431/) (2024)

## Abstract (arXiv)

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.