用三个几何不变量压缩持久拉普拉斯谱，提升学习效率与性能
Analytic Torsion and Spectral Gap Capture Persistent-Laplacian Performance
- 提取贝蒂数、谱间隙和解析扭结三项几何不变量作为紧凑特征
- 在MNIST、QM-3D等数据集上，压缩后性能优于完整谱，且计算量大幅降低
- 适合需要高效拓扑学习的场景，如分子性质预测与图像分类
尽管持久拉普拉斯（PL）比持久同调提供更丰富的几何表征，但其全谱用于学习任务时常受高维性与不同过滤尺度下长度不一问题制约。本文提出一种紧凑的谱表示，将持久拉普拉斯提炼为三个数学上严谨的不变量：贝蒂数、谱间隙与解析扭结。在包括MNIST、QM-3D和SKEMPI WT在内的基准数据集上，该压缩特征空间捕捉了全谱的关键预测信号，某些情况下甚至超越全谱表现，同时显著降低计算开销，并避免高频特征值带来的噪声。结果表明，这些不变量为谱几何与拓扑学习之间提供了原则性强、长度固定的接口。
原文摘要 · Abstract (English)
While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales. We propose a compact spectral representation that distills the persistent Laplacian into three mathematically grounded invariants: Betti numbers, the spectral gap, and analytic torsion. Across benchmark datasets including MNIST, QM-3D, and SKEMPI WT, we demonstrate that this reduced feature space captures the essential predictive signal of the full spectrum, and in some cases outperforms it, while significantly reducing computational overhead and preventing the noise introduced by higher-frequency eigenvalues. Our results suggest that these invariants provide a principled, fixed-length interface between spectral geometry and topological learning.
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。