Research map / Probabilistic and scientific modeling / Molecules and proteins
IsUMap: Manifold Learning and Data Visualization Leveraging Vietoris-Rips Filtrations
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Introduces IsUMap, a manifold learning method that builds local-distortion k-NN star graphs, merges them with a min (canonical t-conorm) gluing, completes distances via Dijkstra and embeds with MDS for visualization.
Problem
Existing manifold learning methods (Isomap, UMAP, LE, t-SNE) struggle with non-uniformly distributed data and local geometric distortions; the paper seeks a principled way to combine local metrics into a global embedding.
Approach
Combine UMAP-style local normalization (scaling distances by per-point kth-neighbor distance) with Isomap-style geodesic completion, framed via Vietoris-Rips filtrations of local star graphs, metric realization of weighted simplicial complexes, and merging by a t-conorm (min on metric weights) before MDS embedding.
Key results
- On MNIST, UMAP gives the best PSI clustering performance, followed by IsUMap, with Isomap worst; all methods peak at dimension 10.
- On hemisphere with non-uniform density, IsUMap with canonical or drastic t-conorm yields a more uniformly distributed disk-like projection than UMAP and Isomap (qualitative).
- Results are qualitative visual comparisons on toy and real datasets; no quantitative table of embedding quality beyond PSI is reported in the text.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use IsUMap instead of Isomap | MNIST · MNIST | Pair Sets Index (PSI) | Helped |
| Use IsUMap instead of UMAP | MNIST · MNIST | Pair Sets Index (PSI) | Hurt |
| Use IsUMap instead of UMAP | non-uniform hemisphere visualization · Hemisphere with non-uniform distribution | geometric visualization quality | Helped |
| Use IsUMap instead of UMAP | trefoil-knotted protein visualization · Trefoil-knotted proteins | underlying structure clarity | Helped |
| Use UMAP instead of IsUMap | trefoil-knotted protein clustering · Trefoil-knotted proteins | cluster separation | Mixed |
| Use canonical t-conorm instead of algebraic sum (ablation) | IsUMap · Hemisphere with non-uniform distribution | projection uniformity | Helped |
| Use canonical t-conorm instead of bounded sum (ablation) | IsUMap · Hemisphere with non-uniform distribution | projection uniformity | Helped |
| Use algebraic sum or bounded sum instead of canonical t-conorm (ablation) | IsUMap · Hemisphere with non-uniform distribution | projection shape | Hurt |
| Remove nearest-neighbor distance subtraction (rho_i) (ablation) | IsUMap · Hemisphere with non-uniform distribution | projection quality | Mixed |
| Increase the number of nearest neighbors (ablation) | IsUMap · Swiss Roll and Swiss Roll with a hole | preservation of intrinsic geometry | Hurt |
| Increase the number of nearest neighbors (ablation) | IsUMap · Torus | embedding quality | No clear effect |
| Increase the number of nearest neighbors (ablation) | IsUMap · MNIST | clustering performance | Helped |
Surprising findings
- UMAP outperforms IsUMap on MNIST clustering, which the authors attribute to UMAP's negative sampling in its embedding step rather than the graph construction.
Limitations
- Quadratic-like complexity makes it slower than UMAP on large datasets.
- Clustering performance on MNIST trails UMAP; pure MDS embedding may be the cause.
- Sensitive to the neighborhood size k on geometric datasets such as Swiss Roll.
Experimental setup
Datasets: Swiss Roll Swiss Roll with a hole Torus Mammoth MNIST Wisconsin breast cancer trefoil-knotted protein chains synthetic hemisphere
Benchmarks: MNIST Wisconsin breast cancer Swiss Roll Torus Mammoth trefoil-knotted protein chains
Metrics: Pair Sets Index (PSI)
Baselines: Isomap UMAP Laplacian Eigenmaps t-SNE
Builds on: UMAP Isomap Laplacian Eigenmaps Spivak metric realization Hausmann theorem
Closest papers by meaning
- Differentiable Mapper for Topological Optimization of Data Representation (ICML 2024 · similarity 0.86)
- Cover learning for large-scale topology representation (ICML 2025 · similarity 0.86)
- Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding (CVPR 2025 · similarity 0.84)
- MANTRA: The Manifold Triangulations Assemblage (ICLR 2025 · similarity 0.84)
- Wasserstein convergence of Cech persistence diagrams for samplings of submanifolds (NeurIPS 2024 · similarity 0.84)
- Point-Level Topological Representation Learning on Point Clouds (ICML 2025 · similarity 0.84)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.