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HOSCF: Efficient decoupling algorithms for finding the best rank-one approximation of higher-order tensors

Chuanfu Xiao, Zeyu Li, Chao Yang

TL;DR

An efficient decoupling algorithm, namely the higher-order self-consistent field (HOSCF) algorithm, inspired by the famous self-consistent field (SCF) iteration, is developed, and an improved HOSCF (iHOSCF) algorithm that incorporates the Rayleigh quotient iteration, which can significantly accelerate the convergence of HOSCF.

Abstract

Best rank-one approximation is one of the most fundamental tasks in tensor computation. In order to fully exploit modern multi-core parallel computers, it is necessary to develop decoupling algorithms for computing the best rank-one approximation of higher-order tensors at large scales. In this paper, we first build a bridge between the rank-one approximation of tensors and the eigenvector-dependent nonlinear eigenvalue problem (NEPv), and then develop an efficient decoupling algorithm, namely the higher-order self-consistent field (HOSCF) algorithm, inspired by the famous self-consistent field (SCF) iteration frequently used in computational chemistry. The convergence theory of the HOSCF algorithm and an estimation of the convergence speed are further presented. In addition, we propose an improved HOSCF (iHOSCF) algorithm that incorporates the Rayleigh quotient iteration, which can significantly accelerate the convergence of HOSCF. Numerical experiments show that the proposed algorithms can efficiently converge to the best rank-one approximation of both synthetic and real-world tensors and can scale with high parallel scalability on a modern parallel computer.

HOSCF: Efficient decoupling algorithms for finding the best rank-one approximation of higher-order tensors

TL;DR

An efficient decoupling algorithm, namely the higher-order self-consistent field (HOSCF) algorithm, inspired by the famous self-consistent field (SCF) iteration, is developed, and an improved HOSCF (iHOSCF) algorithm that incorporates the Rayleigh quotient iteration, which can significantly accelerate the convergence of HOSCF.

Abstract

Best rank-one approximation is one of the most fundamental tasks in tensor computation. In order to fully exploit modern multi-core parallel computers, it is necessary to develop decoupling algorithms for computing the best rank-one approximation of higher-order tensors at large scales. In this paper, we first build a bridge between the rank-one approximation of tensors and the eigenvector-dependent nonlinear eigenvalue problem (NEPv), and then develop an efficient decoupling algorithm, namely the higher-order self-consistent field (HOSCF) algorithm, inspired by the famous self-consistent field (SCF) iteration frequently used in computational chemistry. The convergence theory of the HOSCF algorithm and an estimation of the convergence speed are further presented. In addition, we propose an improved HOSCF (iHOSCF) algorithm that incorporates the Rayleigh quotient iteration, which can significantly accelerate the convergence of HOSCF. Numerical experiments show that the proposed algorithms can efficiently converge to the best rank-one approximation of both synthetic and real-world tensors and can scale with high parallel scalability on a modern parallel computer.
Paper Structure (16 sections, 6 theorems, 71 equations, 3 figures, 5 tables, 1 algorithm)

This paper contains 16 sections, 6 theorems, 71 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A bridge between the rank-one approximation and NEPv
  4. Higher-order self-consistent field (HOSCF)
  5. The algorithm
  6. Computational complexity
  7. Stopping criteria
  8. Improved HOSCF (iHOSCF)
  9. Convergence analysis
  10. Numerical experiments
  11. Convergence test
  12. Example 1
  13. Example 2
  14. Example 3
  15. Parallel scalability
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\bm{x}\in\mathbb{S}^{\sum\limits_{n=1}^{d} I_n-1}$ be a unit vector, the defined NEPv eq:nepv-1 has the following two properties. (P1) If $(\mu;\bm{y})$ is an eigenpair of $\bm{J}(\bm{x})$, then $(-\mu;\hat{\bm{y}})$ is also an eigenpair of $\bm{J}(\hat{\bm{x}})$, where where $\texttt{index}_n$ is the same as Eq. eq:factor. Further, $\mu$ is the largest magnitude eigenvalue of $\bm{J}(\bm{x}

Figures (3)

  • Figure 1: Number of iterations by all tested algorithms for Example 1.
  • Figure 2: The first, third, and fifth texel factors by all tested algorithms for Example 3.
  • Figure 3: Running time (s) and speedup of the HOPM and HOSCF algorithms for tensors of different orders on a multi-core computer.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 4
  • ...and 1 more