Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
On clusterings-good, bad and spectral
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Discrete & Computational Geometry
Generalized Principal Component Analysis (GPCA)
IEEE Transactions on Pattern Analysis and Machine Intelligence
Stability of Persistence Diagrams
Discrete & Computational Geometry
Inferring Local Homology from Sampled Stratified Spaces
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Tighter bounds for random projections of manifolds
Proceedings of the twenty-fourth annual symposium on Computational geometry
Towards a theoretical foundation for Laplacian-based manifold methods
Journal of Computer and System Sciences
Persistent homology for kernels, images, and cokernels
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
A Sampling Theory for Compact Sets in Euclidean Space
Discrete & Computational Geometry
Incremental construction of the delaunay triangulation and the delaunay graph in medium dimension
Proceedings of the twenty-fifth annual symposium on Computational geometry
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Metric graph reconstruction from noisy data
Proceedings of the twenty-seventh annual symposium on Computational geometry
Geometric Inference for Probability Measures
Foundations of Computational Mathematics
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
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The objective of this paper is to show that point cloud data can under certain circumstances be clustered by strata in a plausible way. For our purposes, we consider a stratified space to be a collection of manifolds of different dimensions which are glued together in a locally trivial manner inside some Euclidean space. To adapt this abstract definition to the world of noise, we first define a multi-scale notion of stratified spaces, providing a stratification at different scales which are indexed by a radius parameter. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata. We prove a correctness guarantee for this clustering method under certain topological conditions. We then provide a probabilistic guarantee for the clustering for the point sample setting -- we provide bounds on the minimum number of sample points required to state with high probability which points belong to the same strata. Finally, we give an explicit algorithm for the clustering.