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Persistent Homology for High-dimensional Data Based on Spectral Methods
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Shows that traditional persistent homology fails on high-dimensional noisy point clouds and proposes applying persistent homology to spectral distances (effective resistance, diffusion distance) on the kNN graph, validated on synthetic manifolds and six scRNA-seq datasets.
Problem
Persistent homology is sensitive to high-dimensional noise: when low-intrinsic-dimensional data sit in a high-dimensional ambient space, Euclidean-distance persistence diagrams fail to reveal the true loops and voids.
Approach
Use spectral distances (corrected effective resistance and diffusion distance) computed on the symmetric kNN graph as input distance matrices for persistent homology, since random-walk-based distances aggregate information from all edges and are robust to high-dimensional noise; also derive a closed-form expression of corrected effective resistance as a sum of diffusion distances.
Key results
- On a noisy circle in R^50, Euclidean persistent homology loses its loop at sigma~0.2 while effective resistance and diffusion distance remain robust across noise levels.
- Spectral distances outperformed all competitors on synthetic datasets in R^50 except the torus, where effective resistance matched Euclidean and diffusion performed poorly.
- On six scRNA-seq datasets, spectral methods (especially effective resistance) gave the highest loop detection scores, e.g. both Malaria cycles found with s_2 > 0.9.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace Euclidean distance with effective resistance on the kNN graph | Persistent homology · Noisy circle in R^50 | loop detection score | Helped |
| Replace Euclidean distance with diffusion distance on the kNN graph | Persistent homology · Noisy circle in R^50 | loop detection score | Helped |
| Use spectral distances instead of competing distance methods | Persistent homology · Synthetic datasets in R^50 | hole detection score | Mixed |
| Use diffusion distance instead of Euclidean distance | Persistent homology · Torus in R^50 | hole detection score | Hurt |
| Use effective resistance or diffusion distance instead of Euclidean distance | Persistent homology · Circle with σ=0.25 as ambient dimension increases | loop detection: d≳30 | Helped |
| Replace correlation distance with effective resistance | Persistent homology · Malaria scRNA-seq | hole detection score (s2): s_2 > 0.9 | Helped |
| Replace correlation distance with diffusion distance | Persistent homology · Malaria scRNA-seq | hole detection score (s2): s_2 > 0.9 | Helped |
| Use spectral distances instead of competing distances | Persistent homology · Six scRNA-seq datasets | hole detection score | Helped |
| Use normalized PCA before persistent homology | Persistent homology on 1D toy datasets · Three 1D toy datasets | loop detection score | No clear effect |
| Increase the number of PCA components | PCA preprocessing for persistent homology · 1D toy datasets | loop detection score | Hurt |
| Use fewer PCA components than the true data dimensionality | PCA preprocessing for persistent homology · 1D toy datasets | loop detection score | Hurt |
| Replace PCA preprocessing with effective resistance | Persistent homology · Linked circles and eyeglasses | loop detection score | Helped |
Surprising findings
- Fermat distance and DTM, designed for outlier robustness, gave little benefit or collapsed under high-dimensional Gaussian noise.
- Normalized PCA before persistent homology gave no consistent benefit and required knowing the true dimensionality.
Limitations
- No theoretical proof that spectral distances mitigate the curse of dimensionality
- Persistent homology scales poorly with n and hole dimension; subsampling needed
- Effective resistance and diffusion distances require hyperparameter (k) choice and no principled criterion picks between them
Experimental setup
Datasets: synthetic circle, linked circles, eyeglasses, sphere, torus in R^50 Malaria scRNA-seq Neurosphere scRNA-seq Hippocampus scRNA-seq HeLa2 scRNA-seq Neural IPCs scRNA-seq Pancreas scRNA-seq
Benchmarks: synthetic noisy manifolds in R^50 six single-cell RNA-seq datasets with known cycles
Metrics: hole detection score (relative persistence gap)
Baselines: Euclidean distance persistent homology Fermat distance distance-to-measure (DTM) core distance (HDBSCAN) geodesic distance on kNN graph UMAP graph distance t-SNE graph distance UMAP embedding distance Laplacian Eigenmaps distance PCA preprocessing
Builds on: persistent homology Vietoris-Rips complex diffusion distance effective resistance UMAP
Closest papers by meaning
- The Flood Complex: Large-Scale Persistent Homology on Millions of Points (NeurIPS 2025 · similarity 0.88)
- Graphcode: Learning from multiparameter persistent homology using graph neural networks (NeurIPS 2024 · similarity 0.87)
- Topological Periodicity Test (TopPT) via Confidence Bound of Time-Delay Embeddings (NeurIPS 2026 · similarity 0.87)
- A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams (AAAI 2024 · similarity 0.86)
- Differentiability and Optimization of Multiparameter Persistent Homology (ICML 2024 · similarity 0.86)
- ToMAToMP: Robust and Multi-Parameter Topological Clustering (NeurIPS 2026 · similarity 0.85)
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