# Analytic Torsion and Spectral Gap Capture Persistent-Laplacian Performance

arXiv: https://arxiv.org/abs/2606.16990 · HTML: https://mlsanity.com/paper/2606.16990/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 41 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Stochastic Convergence of Persistence Landscapes and Silhouettes](https://mlsanity.com/paper/1312.0308/) (2013)
- [Persistent Laplacians: properties, algorithms and implications](https://mlsanity.com/paper/2012.02808/) (2020)
- [Representation Geometry as a Diagnostic for Out-of-Distribution Robustness](https://mlsanity.com/paper/2602.03951/) (2026)
- [Noise-Robust Topology Estimation of 2D Image Data via Neural Networks and Persistent Homology](https://mlsanity.com/paper/2509.09140/) (2025)
- [Stable Vectorization of Persistent Laplacians via Spectral Descriptors](https://mlsanity.com/paper/2512.05463/) (2025)
- [Multi-dimensional Persistent Sheaf Laplacians for Image Analysis](https://mlsanity.com/paper/2602.14846/) (2026)

## Abstract (arXiv)

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.