Analytic Torsion and Spectral Gap Capture Persistent-Laplacian Performance
2026-06-15 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address a problem with using persistent Laplacians in data analysis, where the full set of numbers (eigenvalues) can be too large and vary in size, making it hard to use. They propose summarizing this information into three key numbers that still capture the important data shapes. Their tests on various datasets show these summaries work as well or better than using all the numbers, while being simpler and less noisy. This method offers a fixed way to connect complex geometry data to learning tasks.
Persistent LaplacianPersistent HomologyBetti NumbersSpectral GapAnalytic TorsionFiltrationEigenvaluesTopological Data AnalysisMachine LearningSpectral Geometry
Authors
Jernej Grlj, Aaron D. Lauda
Abstract
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.