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ToMAToMP: Robust and Multi-Parameter Topological Clustering

NeurIPS 2026 · Learning theory and algorithms › Clustering · approach: Clustering approximation algorithms · 0 citations (OpenAlex)

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Introduces ToMATopMP, a multi-parameter extension of ToMATo topological clustering that uses MMA decompositions over many diagonal lines to handle several functions at once, removing graph tuning and improving outlier robustness, with theoretical robustness guarantees.

Problem

Topological clustering (ToMATo) handles only one function at a time, needs a user-chosen neighborhood graph, and is sensitive to outlier function values.

Approach

Run ToMATo on a family of diagonal lines in multi-parameter persistence, link ToMATo clusters across lines via MMA decomposition intervals, and assign each point to the most frequently matched cluster; this yields the first multi-function topological clustering with robustness guarantees and corollaries for graph tuning and outliers.

Key results

• On the synthetic dataset, ToMATopMP reaches AMI 0.9256 versus 0.8138 for mean ToMATo across lines and 0.6894 for best k-means variant.
• With injected outliers, ToMATopMP keeps AMI 0.9311 ± 0.0274 on the cell image where ToMATo gets 0.2157 ± 0.1116.
• Running times are 3-4 orders of magnitude longer than baselines (up to ~10^3 s versus ~10^-1 s), a key limitation.

Surprising findings

• Under outliers, spectral clustering did not terminate after a few hours on the tested data.
• For 2-gene rankings on kpmp without outliers, ToMATopMP's Pearson correlation drops to 0.0183, far below ToMATo's ~0.91.

Limitations

• Running time is cubic in graph size and much longer than baselines.
• Robustness theorem relies on assumption (A1) that induced PDs are separated for every line, which is harder as the number of functions grows.
• Gene-pair rankings show low Pearson correlation with ground truth and lower TopHits@10 than ToMATo on some settings.

Experimental setup

Datasets: synthetic 2D dataset (DTM, n=1000) Princeton 3D shape (HKS, n=568) BioImageAnalysisNotebooks cell image (70x70 pixels) kpmp spatial transcriptomics (n=645) spat spatial transcriptomics (n=977)

Benchmarks: synthetic 2D dataset 3D shape segmentation cell image kpmp gene ranking spat gene ranking

Metrics: AMI ARI Pearson correlation TopHits@10

Baselines: ToMATo k-means spectral clustering hierarchical clustering

Builds on: ToMATo MMA decomposition vineyards algorithm AuToMATo multipers

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