Research map / Graph learning / Graph algorithms and clustering
Cover learning for large-scale topology representation
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.
Proposes cover learning, optimizing fuzzy covers via a loss combining measure, boundary, and 0-dimensional persistence terms, and implements it as ShapeDiscover for large-scale topology representation.
Problem
Representing the large-scale topology of geometric datasets with simplicial complexes that are small, geometrically regular, and topologically faithful, where geometric complexes scale poorly and Mapper graphs are hard to tune and only capture low-dimensional information.
Approach
Formulates cover learning as optimization over fuzzy covers (functions into Gamma^{k-1}), with a loss on expected cover measure, boundary measure, and homology-goodness, relaxed to 0-dimensional persistence; the parametric model is a softmax-normalized matrix initialized by spectral clustering, and thresholding the fuzzy cover yields a filtered nerve.
Key results
- ShapeDiscover performs best on the topological inference comparison, followed by Witness (v=1), needing fewer vertices and simplices than prior approaches.
- ShapeDiscover learns the toroidal topology of grid cell population activity with a simplicial complex of just 15 vertices, which Vietoris-Rips cannot achieve with 5000 vertices.
- On human and octopus datasets, ShapeDiscover recovers hollow large-scale topology with few vertices, while 1D Mapper and Differentiable Mapper recover only 1-dimensional structure.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ShapeDiscover instead of Vietoris–Rips | Topological representation of grid-cell population activity · Grid-cell population activity | vertices: 5000 -> 15 | Helped |
| Use ShapeDiscover instead of standard topological inference approaches | Topological inference · Synthetic 2-sphere, synthetic 3-sphere, human surface, and dynamical-system video data | vertices needed for homology recovery | Helped |
| Use ShapeDiscover instead of 1D Mapper or Differentiable Mapper | Large-scale topology visualization · Human and octopus datasets | topological information represented | Helped |
| Use ShapeDiscover instead of Ball Mapper | Large-scale topology visualization · Human and octopus datasets | topology recovery | Helped |
| Remove the geometry loss (ablation) | ShapeDiscover · Experiments reported in the paper | experimental results | No clear effect |
Surprising findings
- The geometry loss term plays little role in experiments; results are very similar with or without it, attributed to the L2 regularization.
Limitations
- Covers can contain spurious intersections, complicating visualization and topology recovery.
- No theoretical consistency or topology-recovery guarantees for optimization-based cover learning.
- Topological persistence optimization is slow and cannot use mini-batches.
Experimental setup
Datasets: synthetic 2-sphere synthetic 3-sphere human surface point cloud dynamical system video data (2-torus) grid cell population activity human and octopus datasets small handwritten digits MNIST C. elegans gene expression
Benchmarks: synthetic 2-sphere synthetic 3-sphere human surface torus video data MNIST small digits C. elegans
Metrics: homology recovery quotient number of vertices needed total number of simplices
Baselines: 1D Mapper Ball Mapper Differentiable Mapper Vietoris-Rips complex Witness complex UMAP
Builds on: Mapper Ball Mapper Differentiable Mapper UMAP topological persistence optimization with big steps
Closest papers by meaning
- Differentiable Mapper for Topological Optimization of Data Representation (ICML 2024 · similarity 0.89)
- Point-Level Topological Representation Learning on Point Clouds (ICML 2025 · similarity 0.87)
- Simplicial Representation Learning with Neural $k$-Forms (ICLR 2024 · similarity 0.86)
- IsUMap: Manifold Learning and Data Visualization Leveraging Vietoris-Rips Filtrations (AAAI 2025 · similarity 0.86)
- Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity (ICLR 2025 · similarity 0.85)
- Geometry Awakening: Cross-Geometry Learning Exhibits Superiority over Individual Structures (NeurIPS 2024 · similarity 0.85)
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.