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Differentiable Mapper for Topological Optimization of Data Representation

ICML 2024 · Graph learning › Graph algorithms and clustering · approach: Graph distance and retrieval · 1 citations (OpenAlex)

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Introduces Soft Mapper, a differentiable probabilistic relaxation of Mapper graphs, and uses it to optimize linear filter functions by gradient descent on persistence-based topological losses, with convergence guarantees.

Problem

Mapper graphs are highly sensitive to their parameters, and the filter function has never been tuned automatically because it is a general continuous function with a wild parameter space; the standard combinatorial Mapper is also non-differentiable in the filter parameters.

Approach

Generalize Mapper into a distribution over Mapper graphs via Bernoulli cover assignments, with a smooth relaxation of interval membership (bump-like functions of width delta) so the expected topological loss (total extended persistence) is smooth and definable in an o-minimal structure, enabling SGD convergence guarantees for filter parameters.

Key results

• Optimized linear filters recover intuitive directions on 3D shapes with correlations 0.9999 (human), -0.9984 (octopus), 0.9993 (table).
• On human preimplantation data, Pearson correlation of filter with sampling time improves from 0.1330 (initial) to -0.7549 (optimized).
• On MEF reprogramming data, correlation with time improves from -0.0560 to -0.4015.

Measured effects reported in the paper

Surprising findings

• The optimized direction for the table is different from the first principal components of PCA, favoring a surface-perpendicular filter over maximal variance.

Limitations

• Only linear filter families are optimized in experiments; non-linear filters are left to future work.
• Results are proof-of-concept and not state-of-the-art representations for the scRNA-seq domain.
• Convergence guarantees are only to critical points, and the standard (non-relaxed) cover assignment violates the smoothness assumption.

Experimental setup

Datasets: 3D shape point clouds (human, octopus, table) human preimplantation embryo scRNA-seq (1,529 cells) mouse embryonic fibroblasts reprogramming scRNA-seq (subsample of 1,500 of 251,203 cells)

Benchmarks: 3D shape skeletonization qualitative and direction correlation human preimplantation embryo time correlation MEF reprogramming time correlation

Metrics: correlation between optimized and intuitive direction Pearson correlation with sampling time learning curves

Baselines: initial (arbitrary) filter functions PCA directions

Builds on: Mapper algorithm extended persistence o-minimal stochastic gradient descent convergence (Davis et al.)

Closest papers by meaning

• Cover learning for large-scale topology representation (ICML 2025 · similarity 0.89)
• Diffeomorphic interpolation for efficient persistence-based topological optimization (NeurIPS 2024 · similarity 0.86)
• IsUMap: Manifold Learning and Data Visualization Leveraging Vietoris-Rips Filtrations (AAAI 2025 · similarity 0.86)
• ToMAToMP: Robust and Multi-Parameter Topological Clustering (NeurIPS 2026 · similarity 0.86)
• Point-Level Topological Representation Learning on Point Clouds (ICML 2025 · similarity 0.85)
• TopER: Topological Embeddings in Graph Representation Learning (NeurIPS 2025 · similarity 0.85)