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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.16130 |
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| _version_ | 1866909469865148416 |
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| author | Harb, Elfarouk Lam, Ho Shan |
| author_facet | Harb, Elfarouk Lam, Ho Shan |
| contents | Efficiently solving sparse linear systems $Ax=b$, where $A$ is a large, sparse, symmetric positive semi-definite matrix, is a core challenge in scientific computing, machine learning, and optimization. A major bottleneck in Gaussian elimination for these systems is fill-in, the creation of non-zero entries that increase memory and computational cost. Minimizing fill-in is NP-hard, and existing heuristics like Minimum Degree and Nested Dissection offer limited adaptability across diverse problem instances.
We introduce \textit{ReFill}, a reinforcement learning framework enhanced by Graph Neural Networks (GNNs) to learn adaptive ordering strategies for fill-in minimization. ReFill trains a GNN-based heuristic to predict efficient elimination orders, outperforming traditional heuristics by dynamically adapting to the structure of input matrices. Experiments demonstrate that ReFill outperforms strong heuristics in reducing fill-in, highlighting the untapped potential of learning-based methods for this well-studied classical problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16130 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | ReFill: Reinforcement Learning for Fill-In Minimization Harb, Elfarouk Lam, Ho Shan Machine Learning Efficiently solving sparse linear systems $Ax=b$, where $A$ is a large, sparse, symmetric positive semi-definite matrix, is a core challenge in scientific computing, machine learning, and optimization. A major bottleneck in Gaussian elimination for these systems is fill-in, the creation of non-zero entries that increase memory and computational cost. Minimizing fill-in is NP-hard, and existing heuristics like Minimum Degree and Nested Dissection offer limited adaptability across diverse problem instances. We introduce \textit{ReFill}, a reinforcement learning framework enhanced by Graph Neural Networks (GNNs) to learn adaptive ordering strategies for fill-in minimization. ReFill trains a GNN-based heuristic to predict efficient elimination orders, outperforming traditional heuristics by dynamically adapting to the structure of input matrices. Experiments demonstrate that ReFill outperforms strong heuristics in reducing fill-in, highlighting the untapped potential of learning-based methods for this well-studied classical problem. |
| title | ReFill: Reinforcement Learning for Fill-In Minimization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2501.16130 |