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A Novel Block-Alternating Iterative Algorithm for Retrieving Top-$k$ Elements from Factorized Tensors

Chuanfu Xiao, Jiaxin Zeng

TL;DR

This work addresses retrieving the top-$k$ elements from factorized tensors, specifically CP-tensor representations, by reformulating the problem as a continuous constrained optimization grounded in a symmetric eigenvalue framework. It introduces a novel block-alternating iterative algorithm that reduces a large optimization into small, tractable subproblems, aided by a heuristic that exploits separable objective structure. The approach shows superior accuracy and stability compared to baselines across synthetic CP tensors, tensors derived from multivariate functions, and quantum circuit simulations. The results indicate robust performance for both $k=1$ and $k>1$, suggesting broad applicability to large-scale tensor data in areas such as recommendation systems and quantum simulations.

Abstract

Tensors, especially higher-order tensors, are typically represented in low-rank formats to preserve the main information of the high-dimensional data while saving memory space. In practice, only a small fraction elements in high-dimensional data are of interest, such as the $k$ largest or smallest elements. Thus, retrieving the $k$ largest/smallest elements from a low-rank tensor is a fundamental and important task in a wide variety of applications. In this paper, we first model the top-$k$ elements retrieval problem to a continuous constrained optimization problem. To address the equivalent optimization problem, we develop a block-alternating iterative algorithm that decomposes the original problem into a sequence of small-scale subproblems. Leveraging the separable summation structure of the objective function, a heuristic algorithm is proposed to solve these subproblems in an alternating manner. Numerical experiments with tensors from synthetic and real-world applications demonstrate that the proposed algorithm outperforms existing methods in terms of accuracy and stability.

A Novel Block-Alternating Iterative Algorithm for Retrieving Top-$k$ Elements from Factorized Tensors

TL;DR

This work addresses retrieving the top- elements from factorized tensors, specifically CP-tensor representations, by reformulating the problem as a continuous constrained optimization grounded in a symmetric eigenvalue framework. It introduces a novel block-alternating iterative algorithm that reduces a large optimization into small, tractable subproblems, aided by a heuristic that exploits separable objective structure. The approach shows superior accuracy and stability compared to baselines across synthetic CP tensors, tensors derived from multivariate functions, and quantum circuit simulations. The results indicate robust performance for both and , suggesting broad applicability to large-scale tensor data in areas such as recommendation systems and quantum simulations.

Abstract

Tensors, especially higher-order tensors, are typically represented in low-rank formats to preserve the main information of the high-dimensional data while saving memory space. In practice, only a small fraction elements in high-dimensional data are of interest, such as the largest or smallest elements. Thus, retrieving the largest/smallest elements from a low-rank tensor is a fundamental and important task in a wide variety of applications. In this paper, we first model the top- elements retrieval problem to a continuous constrained optimization problem. To address the equivalent optimization problem, we develop a block-alternating iterative algorithm that decomposes the original problem into a sequence of small-scale subproblems. Leveraging the separable summation structure of the objective function, a heuristic algorithm is proposed to solve these subproblems in an alternating manner. Numerical experiments with tensors from synthetic and real-world applications demonstrate that the proposed algorithm outperforms existing methods in terms of accuracy and stability.

Paper Structure

This paper contains 10 sections, 2 theorems, 21 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. algorithm
  4. The block-alternating iterative algorithm
  5. Complexity analysis
  6. Numerical experiments
  7. Tests on random CP tensors
  8. Tests on CP tensors from multivariate functions
  9. Tests on quantum circuit simulations
  10. Conclusion

Key Result

Theorem 1

Let $\bm{\mathcal{A}}$ be a $d$th-order CP tensor with its CP representation given by Eq. eq:cp. Subsequently, retrieving the $k$ largest elements of $\bm{\mathcal{A}}$ is equivalent to solving the following continuous constrained optimization problem:

Figures (9)

  • Figure 1: The diagram of a third-order CP tensor.
  • Figure 2: The workflow of the block-alternating iterative algorithm for the top-$k$ elements retrieval problem in factorized tensors.
  • Figure 3: The largest value and accuracy obtained by baselines and our algorithm for random CP tensors with different uniform distributions.
  • Figure 4: The smallest value and accuracy obtained by baselines and our algorithm for random CP tensors with different uniform distributions.
  • Figure 5: The sum of the 5 largest values and accuracy obtained by baselines and our algorithm for random CP tensors with different uniform distributions.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • Remark 4