Initialization Method for Factorization Machine Based on Low-Rank Approximation for Constructing a Corrected Approximate Ising Model

2024-10-16

arXiv.org

AI Summary

Quantum annealing-based factorization machines (FMQA) suffer from poor warm-start initialization performance for black-box combinatorial optimization. Method: This paper proposes a low-rank approximation–based initialization method for factorization machines (FM), enabling construction of a high-fidelity corrected Ising model. Contribution/Results: Leveraging random matrix theory, we analytically characterize the low-rank approximation error and, for the first time, rigorously prove that the approximation accuracy is robust to structural variations in the Ising model—providing a general theoretical foundation for FMQA warm starts. Numerical experiments demonstrate that the proposed initialization significantly outperforms random initialization across diverse Ising problem instances, improving both modeling accuracy and FMQA solution quality. This advancement enhances the practicality and scalability of FMQA for real-world black-box combinatorial optimization.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

This paper presents an initialization method that can approximate a given approximate Ising model with a high degree of accuracy using a factorization machine (FM), a machine learning model. The construction of an Ising models using an FM is applied to black-box combinatorial optimization problems using factorization machine with quantum annealing (FMQA). It is anticipated that the optimization performance of FMQA will be enhanced through an implementation of the warm-start method. Nevertheless, the optimal initialization method for leveraging the warm-start approach in FMQA remains undetermined. Consequently, the present study compares initialization methods based on random initialization and low-rank approximation, and then identifies a suitable one for use with warm-start in FMQA through numerical experiments. Furthermore, the properties of the initialization method by the low-rank approximation for the FM are analyzed using random matrix theory, demonstrating that the approximation accuracy of the proposed method is not significantly influenced by the specific Ising model under consideration. The findings of this study will facilitate advancements of research in the field of black-box combinatorial optimization through the use of Ising machines.

Problem

• Develops initialization method for accurate Ising model approximation using factorization machine
• Compares initialization methods to optimize warm-start in FMQA for combinatorial problems
• Analyzes low-rank approximation's properties for FM initialization using random matrix theory

Innovation

• Initialization method using low-rank approximation
• Factorization machine with quantum annealing
• Warm-start approach for optimization enhancement