COMPLEX：多参数持久性模的闭式认证嵌入
COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules
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中文总结 AI 辅助
COMPLEX提出闭式无训练的多参数持久性模嵌入，首次提供双侧失真界，使特征保真可度量，并在多个基准上达到最先进性能。
中文摘要 AI 辅助
我们所知的所有多参数持久性向量化方法都带有单侧Lipschitz上界，而没有任何下界：没有下界度量，特征就没有保真性可言，也无法在此基础上构建任何逐预测保证。本文补上了缺失的这一侧。COMPLEX是一种闭式、无需训练的嵌入方法，用于多参数模——沿固定的近对角网对模进行切片，通过认证的PLACE/PALACE地标映射对每个切片条形码进行嵌入，然后拼接。在可检验的见证切片一致性条件下（该条件在Orbit5k上100%的审计对中成立），单个切片携带闭式下界度量：分离的模在嵌入中保持分离。结合标准上界，这据我们所知首次为多参数特征映射提供了双侧失真界，使保真性可度量。通过度量，我们发现下界在实际距离的小因子内是紧的，但在操作上是局部的：在相同特征上，RBF-SVM达到91%，而1-NN达到78%。因此，逐预测的局部认证因结构原因而失败，这普遍存在于每个由单一坐标见证下界度量的地标嵌入中。无需学习嵌入，也无需保留校准集——仅使用交叉验证的SVM分类头——COMPLEX在两个Orbit基准上达到了最先进水平（Orbit5k上91.95%，Orbit100k上92.98%），与Euler特征曲面持平或更高，并优于transformer和graphcode。在图数据上，它以单一固定配置在全部四个共享分子基准上超过GRIL，包括唯一在COX2上以超过三个百分点超越多数类基线的多参数方法。闭式选择——地标半径、核（保持证书）和双过滤集——可进一步提高精度；而基于梯度的自适应则无增益。
英文摘要
Every multiparameter persistence vectorization we know of carries a one-sided Lipschitz upper bound and nothing below it: without a lower gauge there is no sense in which the features are faithful, and no per-prediction guarantee can be built on them. This paper supplies the missing side. COMPLEX is a closed-form, training-free embedding of multiparameter modules -- slice the module along a fixed near-diagonal net, embed each slice barcode by the certified PLACE/PALACE landmark map, concatenate. Under a checkable witnessing-slice coherence condition, holding on 100% of audited pairs on Orbit5k, a single slice carries a closed-form lower gauge: separated modules stay separated in the embedding. With the standard upper bound this gives, to our knowledge, the first two-sided distortion bound for a multiparameter feature map, making faithfulness measurable. Measuring it, we find the floor tight within a small factor of realized distances yet operationally local: an RBF-SVM reaches 91% where 1-NN reaches 78% on the same features. Local per-prediction certification therefore fails for a structural reason common to every landmark embedding whose lower gauge is witnessed by one coordinate. With no learned embedding and no held-out calibration -- only a cross-validated SVM head -- COMPLEX sets the state of the art on both Orbit benchmarks (91.95% on Orbit5k, 92.98% on Orbit100k), level with or above Euler-characteristic surfaces and above transformers and graphcode. On graphs it exceeds GRIL on all four shared molecular benchmarks with one fixed configuration, including the only multiparameter method to clear COX2's majority baseline by more than three points. Closed-form selection -- of the landmark radius, the kernel (certificate-preserving), and the bifiltration set -- buys further accuracy; gradient-shaped adaptation buys none.
发表机构
- George Washington University（乔治华盛顿大学）
- Montana Technological University（蒙大拿科技大学）
- University of Ljubljana（卢布尔雅那大学）
- Institute IMFM（IMFM 研究所）
机构由 AI 辅助整理，请以论文原文为准。
