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

TL;DR

This paper introduces a new block-alternating iterative algorithm designed to efficiently retrieve the top-$k$ elements from low-rank factorized tensors, addressing a key computational challenge in high-dimensional data analysis.

Contribution

The paper proposes a novel block-alternating iterative method that decomposes the top-$k$ element retrieval into manageable subproblems, improving accuracy and stability over existing approaches.

Findings

Outperforms existing methods in accuracy

Demonstrates superior stability in numerical experiments

Effective on both synthetic and real-world tensors

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.