Matrix Phylogeny: Compact Spectral Fingerprints for Trap-Robust Preconditioner Selection

TL;DR

This paper introduces compact spectral fingerprints derived from matrix properties that enable efficient, robust, and accurate clustering and preconditioner selection without eigendecomposition, suitable for large matrix analysis.

Contribution

It proposes low-dimensional, eigendecomposition-free spectral descriptors (CSF/ASF) that are permutation and scale invariant, improving matrix classification and preconditioner selection.

Findings

CSF with K=3-5 achieves perfect clustering on synthetic datasets.

Both CSF-H and ASF-H reach ARI=1.0 on the SuiteSparse benchmark.

Descriptors are stable to noise and outperform alternatives in accuracy and efficiency.

Abstract

Matrix Phylogeny introduces compact spectral fingerprints (CSF/ASF) that characterize matrices at the family level. These fingerprints are low-dimensional, eigendecomposition-free descriptors built from Chebyshev trace moments estimated by Hutchinson sketches. A simple affine rescaling to [-1,1] makes them permutation/similarity invariant and robust to global scaling. Across synthetic and real tests, we observe phylogenetic compactness: only a few moments are needed. CSF with K=3-5 already yields perfect clustering (ARI=1.0; silhouettes ~0.89) on four synthetic families and a five-family set including BA vs ER, while ASF adapts the dimension on demand (median K*~9). On a SuiteSparse mini-benchmark (Hutchinson p~100), both CSF-H and ASF-H reach ARI=1.0. Against strong alternatives (eigenvalue histograms + Wasserstein, heat-kernel traces, WL-subtree), CSF-K=5 matches or exceeds accuracy while avoiding eigendecompositions and using far fewer features (K<=10 vs 64/9153). The descriptors are stable to noise (log-log slope ~1.03, R^2~0.993) and support a practical trap->recommend pipeline for automated preconditioner selection. In an adversarial E6+ setting with a probe-and-switch mechanism, our physics-guided recommender attains near-oracle iteration counts (p90 regret=0), whereas a Frobenius 1-NN baseline exhibits large spikes (p90~34-60). CSF/ASF deliver compact (K<=10), fast, invariant fingerprints that enable scalable, structure-aware search and recommendation over large matrix repositories. We recommend CSF with K=5 by default, and ASF when domain-specific adaptivity is desired.