# Persistent Homology for High-dimensional Data Based on Spectral Methods

arXiv: https://arxiv.org/abs/2311.03087 · HTML: https://mlsanity.com/paper/2311.03087/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 80 references that are in the map.
- First posted on arXiv 6 Nov 2023.

## Builds on (most cited first, 22 of 22)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Kernel Operations on the GPU, with Autodiff, without Memory Overflows](https://mlsanity.com/paper/2004.11127/) (2020)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [Attraction-Repulsion Spectrum in Neighbor Embeddings](https://mlsanity.com/paper/2007.08902/) (2020)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [Persistence Homology of Networks: Methods and Applications](https://mlsanity.com/paper/1907.08708/) (2019)
- [Persistent Laplacians: properties, algorithms and implications](https://mlsanity.com/paper/2012.02808/) (2020)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Subsampling Methods for Persistent Homology](https://mlsanity.com/paper/1406.1901/) (2014)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)
- [Coarse Graining of Data via Inhomogeneous Diffusion Condensation](https://mlsanity.com/paper/1907.04463/) (2019)
- [Intrinsic persistent homology via density-based metric learning](https://mlsanity.com/paper/2012.07621/) (2020)
- [The Manifold Scattering Transform for High-Dimensional Point Cloud Data](https://mlsanity.com/paper/2206.10078/) (2022)
- [Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology](https://mlsanity.com/paper/1707.06683/) (2017)
- [Robust Persistence Diagrams using Reproducing Kernels](https://mlsanity.com/paper/2006.10012/) (2020)
- [Generic families of finite metric spaces with identical or trivial 1-dimensional persistence](https://mlsanity.com/paper/2202.00577/) (2022)
- [Persistent Homology of Coarse Grained State Space Networks](https://mlsanity.com/paper/2206.02530/) (2022)

## Built on by (most cited first, 3 of 3)

- [Manifold Approximation leads to Robust Kernel Alignment](https://mlsanity.com/paper/2510.22953/) (2025)
- [Beyond Explained Variance: A Cautionary Tale of PCA](https://mlsanity.com/paper/2605.13520/) (2026)
- [HyperShadow: A Benchmark for Detecting 3D Projections of Higher-Dimensional Spatial Objects](https://mlsanity.com/paper/2607.14419/) (2026)

## Abstract (arXiv)

Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much higher dimensionality. We show that in this case traditional persistent homology becomes very sensitive to noise and fails to detect the correct topology. The same holds true for existing refinements of persistent homology. As a remedy, we find that spectral distances on the k-nearest-neighbor graph of the data, such as diffusion distance and effective resistance, allow to detect the correct topology even in the presence of high-dimensional noise. Moreover, we derive a novel closed-form formula for effective resistance, and describe its relation to diffusion distances. Finally, we apply these methods to high-dimensional single-cell RNA-sequencing data and show that spectral distances allow robust detection of cell cycle loops.