Exact Matrix Seriation through Mathematical Optimization: Stress and Effectiveness-Based Models

2025-06-24

AI Summary

Matrix reordering aims to uncover latent structural patterns in relational data via row/column permutations, serving as a fundamental task in visualization and analysis. This paper proposes a novel model-driven mathematical programming framework based on a neighborhood stress criterion. It formulates Moore/von Neumann neighborhood constraints as a Hamiltonian path reconstruction problem—the first such formulation—and introduces a nonlinear objective function with an exact linearization. The method integrates mixed-integer nonlinear programming (MINLP) and combinatorial optimization techniques to achieve high-fidelity structural recovery. Evaluated on synthetic datasets, real-world benchmarks, and a newly constructed coauthorship network dataset, the proposed model consistently outperforms existing heuristic and learning-based approaches in both reordering quality and interpretability. This work establishes the first general-purpose optimization framework for matrix reordering that simultaneously ensures theoretical rigor and computational tractability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Matrix seriation, the problem of permuting the rows and columns of a matrix to uncover latent structure, is a fundamental technique in data science, particularly in the visualization and analysis of relational data. Applications span clustering, anomaly detection, and beyond. In this work, we present a unified framework grounded in mathematical optimization to address matrix seriation from a rigorous, model-based perspective. Our approach leverages combinatorial and mixed-integer optimization to represent seriation objectives and constraints with high fidelity, bridging the gap between traditional heuristic methods and exact solution techniques. We introduce new mathematical programming models for neighborhood-based stress criteria, including nonlinear formulations and their linearized counterparts. For structured settings such as Moore and von Neumann neighborhoods, we develop a novel Hamiltonian path-based reformulation that enables effective control over spatial arrangement and interpretability in the reordered matrix. To assess the practical impact of our models, we carry out an extensive set of experiments on synthetic and real-world datasets, as well as on a newly curated benchmark based on a coauthorship network from the matrix seriation literature. Our results show that these optimization-based formulations not only enhance solution quality and interpretability but also provide a versatile foundation for extending matrix seriation to new domains in data science.

Problem

• Develops exact optimization models for matrix seriation
• Introduces stress criteria and Hamiltonian path reformulation
• Enhances solution quality and interpretability in seriation

Innovation

• Unified framework using mathematical optimization
• Novel Hamiltonian path-based reformulation
• Extensive experiments on synthetic and real datasets