用少量特征精准识别矩阵家族，助力自动选预条件器。
Matrix Phylogeny: Compact Spectral Fingerprints for Trap-Robust Preconditioner Selection
- 基于切比雪夫迹矩的低维指纹，免去特征值分解。
- 仅需3-5个矩就实现完美聚类（ARI=1.0），抗噪声稳定。
- 适合大规模矩阵库中结构感知的自动化推荐场景。
Matrix Phylogeny 提出紧凑谱指纹（CSF/ASF），以低维、无需特征值分解的方式刻画矩阵的家族级特性。这些指纹通过哈钦森采样估计切比雪夫迹矩构建，经[-1,1]仿射缩放后具备置换/相似性不变性及全局缩放鲁棒性。在合成与真实测试中均表现出谱系紧凑性：仅需3-5个矩即在四类合成家族和含BA vs ER的五家族集合上实现完美聚类（ARI=1.0；轮廓系数~0.89）。在SuiteSparse小型基准上（哈钦森采样次数p≈100），CSF-H与ASF-H均达ARI=1.0。相比强基线方法（特征值直方图+沃瑟斯坦距离、热核迹、WL子树），CSF-K=5在避免特征值分解的同时，仅用不超过10个特征（对比64/9153）即匹配或超越精度。指纹对噪声稳定（对数-对数斜率~1.03，R²~0.993），支持实用的陷阱→推荐流水线。在对抗性E6+设置下，物理引导推荐器逼近最优迭代次数（p90后悔=0），而弗罗贝尼乌斯1-NN基线出现显著波动（p90~34-60）。CSF/ASF提供紧凑（K≤10）、快速、不变的指纹，支持大规模矩阵库中的可扩展、结构感知搜索与推荐。默认推荐使用CSF-K=5，领域自适应场景下建议采用ASF。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。