Factorization-in-Loop: Proximal Fill-in Minimization for Sparse Matrix Reordering

2025-11-12

AI Summary

Fill-in elements in sparse matrix LU factorization significantly increase memory and computational overhead; however, minimizing fill-in via reordering is NP-hard, and existing surrogate objectives lack theoretical guarantees. This paper introduces the first end-to-end differentiable reordering framework that explicitly embeds the LU factorization process into the learning objective. We model matrix structure using a graph encoder and propose two reparameterization strategies to bridge discrete permutations and continuous optimization. Our method integrates ℓ₁ regularization, the alternating direction method of multipliers (ADMM), and proximal gradient descent. Theoretically, it improves consistency in approximating fill-in minimization. Evaluated on the SuiteSparse benchmark, our approach reduces fill-in by 20% and LU factorization time by 17.8% on average compared to state-of-the-art methods, substantially enhancing sparse linear solver efficiency.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/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: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Fill-ins are new nonzero elements in the summation of the upper and lower triangular factors generated during LU factorization. For large sparse matrices, they will increase the memory usage and computational time, and be reduced through proper row or column arrangement, namely matrix reordering. Finding a row or column permutation with the minimal fill-ins is NP-hard, and surrogate objectives are designed to derive fill-in reduction permutations or learn a reordering function. However, there is no theoretical guarantee between the golden criterion and these surrogate objectives. Here we propose to learn a reordering network by minimizing (l_1) norm of triangular factors of the reordered matrix to approximate the exact number of fill-ins. The reordering network utilizes a graph encoder to predict row or column node scores. For inference, it is easy and fast to derive the permutation from sorting algorithms for matrices. For gradient based optimization, there is a large gap between the predicted node scores and resultant triangular factors in the optimization objective. To bridge the gap, we first design two reparameterization techniques to obtain the permutation matrix from node scores. The matrix is reordered by multiplying the permutation matrix. Then we introduce the factorization process into the objective function to arrive at target triangular factors. The overall objective function is optimized with the alternating direction method of multipliers and proximal gradient descent. Experimental results on benchmark sparse matrix collection SuiteSparse show the fill-in number and LU factorization time reduction of our proposed method is 20% and 17.8% compared with state-of-the-art baselines.

Problem

• Reducing fill-ins during LU factorization of sparse matrices
• Learning reordering networks to minimize triangular factor norms
• Bridging gap between node scores and factorization objectives

Innovation

• Minimizes l1 norm of triangular factors
• Uses graph encoder to predict node scores
• Employs ADMM and proximal gradient descent