Relative-Absolute Fusion: Rethinking Feature Extraction in Image-Based Iterative Method Selection for Solving Sparse Linear Systems

TL;DR

This work targets robust selection of iterative solvers for sparse linear systems by addressing feature ambiguity in image-based representations. It introduces Relative-Absolute Fusion (RAF), a dual-feature extraction that jointly leverages relative image features and absolute numerical values to fully characterize matrices, integrated into an image-based method selection pipeline. To support learning, the authors create BMCMat, a balanced PDE-discretized dataset that mitigates SuiteSparse’s class imbalance. Empirical results on SuiteSparse and BMCMat show RAF achieving state-of-the-art selection accuracy and reducing solution times by up to 0.79 seconds, demonstrating practical gains in speed and reliability for solving sparse systems.

Abstract

Iterative method selection is crucial for solving sparse linear systems because these methods inherently lack robustness. Though image-based selection approaches have shown promise, their feature extraction techniques might encode distinct matrices into identical image representations, leading to the same selection and suboptimal method. In this paper, we introduce RAF (Relative-Absolute Fusion), an efficient feature extraction technique to enhance image-based selection approaches. By simultaneously extracting and fusing image representations as relative features with corresponding numerical values as absolute features, RAF achieves comprehensive matrix representations that prevent feature ambiguity across distinct matrices, thus improving selection accuracy and unlocking the potential of image-based selection approaches. We conducted comprehensive evaluations of RAF on SuiteSparse and our developed BMCMat (Balanced Multi-Classification Matrix dataset), demonstrating solution time reductions of 0.08s-0.29s for sparse linear systems, which is 5.86%-11.50% faster than conventional image-based selection approaches and achieves state-of-the-art (SOTA) performance. BMCMat is available at https://github.com/zkqq/BMCMat.

Figures (7)

• Figure 1: Motivation for RAF. Conventional feature extraction techniques in image-based selection approaches encode distinct matrices $\mathcal{A}_1$ and $\mathcal{A}_2$ into identical RGB image representations, yielding the same selection method (CG & SSOR). Although the CG & SSOR method efficiently solves $\mathcal{A}_1x=b$, it exhibits poor performance for $\mathcal{A}_2x=b$.
• Figure 2: Distribution of optimal iterative methods across datasets. For clarity, only seven methods with the highest percentages are listed for the SuiteSparse dataset.
• Figure 3: Comparison between conventional feature extraction techniques (left) and RAF (right) in image-based selection approaches. RAF simultaneously extracts relative image representations and corresponding absolute numerical values, yielding comprehensive matrix features for enhanced characterization. Notably, because $N_{\mathcal{A}}$ demonstrates higher effectiveness than the blue channel, RAF retains only the red and green channels in the image representation.
• Figure 4: Pipeline for image-based iterative method selection with RAF. Component (a) extracts the red and green channels as relative matrix features, which are subsequently learned via convolution to capture spatial patterns within the matrix. Component (b) simultaneously extracts six corresponding numerical values as absolute matrix features and learns them through linear layers. Finally, component (c) fuses these features to select the optimal iterative method for the given matrix.
• Figure 5: Selection accuracy of FC, CNN, GNN, and RAF on SuiteSparse and BMCMat.