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.

翻译：尽管持续拉普拉斯算子（PL）比持续同调提供了更丰富的数据几何表示，但利用其完整特征谱进行学习任务时，常受限于高维度以及不同过滤尺度下的“变长”问题。我们提出了一种紧凑的谱表示方法，将持续拉普拉斯算子提炼为三个数学基础不变量：贝蒂数、谱间隙和解析挠率。在包含MNIST、QM-3D和SKEMPI WT的基准数据集上，我们证明这种降维特征空间能够捕获完整谱的本质预测信号，在某些情况下甚至超越后者，同时显著降低计算开销并防止高频特征值引入的噪声。我们的结果表明，这些不变量为谱几何与拓扑学习之间提供了原理性的定长接口。