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The Rank-1 Completion Problem for Cubic Tensors

Jinling Zhou, Jiawang Nie, Zheng Peng, Guangming Zhou

TL;DR

This paper shows that this problem is equivalent to a special rank-$1$ matrix recovery problem and it can be solved by an iterative formula, and proposes both nuclear norm relaxation and moment relaxation methods for solving the resulting rank-$1$ matrix recovery problem.

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.

The Rank-1 Completion Problem for Cubic Tensors

TL;DR

This paper shows that this problem is equivalent to a special rank- matrix recovery problem and it can be solved by an iterative formula, and proposes both nuclear norm relaxation and moment relaxation methods for solving the resulting rank- matrix recovery problem.

Abstract

This paper studies the rank- tensor completion problem for cubic tensors. First of all, we show that this problem is equivalent to a special rank- matrix recovery problem. When the tensor is strongly rank- completable, we show that the problem is equivalent to a rank- 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- matrix recovery problem. The nuclear norm relaxation sometimes returns a rank- tensor completion, while sometimes it does not. When it fails, we apply the moment hierarchy of semidefinite programming relaxations to solve the rank- matrix recovery problem. The moment hierarchy can always get a rank- tensor completion, or detect its nonexistence. Numerical experiments are shown to demonstrate the efficiency of these proposed methods.
Paper Structure (16 sections, 2 theorems, 107 equations, 4 tables)

This paper contains 16 sections, 2 theorems, 107 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Moment and localizing matrices
  5. Rank-1 tensor completion and matrix recovery
  6. Reduction to matrix recovery
  7. Strongly rank-1 completable tensors
  8. Nuclear norm relaxation
  9. The Moment-SOS Relaxations
  10. Symmetric rank-1 tensor completions
  11. Extension to higher order tensors
  12. Numerical experiments
  13. Performance of nuclear norm relaxations
  14. Performance for strongly rank-1 completable tensors
  15. Performance of moment relaxations
  16. ...and 1 more sections

Key Result

Theorem 3.1

Suppose $\mathcal{A}$ is the partially given tensor as above. If $X=ab^T$ is feasible for (eq-equivalence) and for each $k =1, \ldots, n_3$, the following equation has a nonzero coefficient for $c_k$, then $\mathcal{A}$ has the rank-$1$ completion $\mathcal{A} = a \otimes b \otimes c$ with $c =(c_1, \ldots, c_{n_3})$. Conversely, if $\mathcal{A}$ is rank-$1$ completable and $\mathcal{A}_{\hat{i}

Theorems & Definitions (15)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Definition 3.3
  • Example 3.4
  • Example 3.5
  • Theorem 4.1
  • proof
  • Example 6.1
  • Example 7.1
  • ...and 5 more