Hierarchical morse complexes for piecewise linear 2-manifolds
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Introduction to algorithms
Mean Shift: A Robust Approach Toward Feature Space Analysis
IEEE Transactions on Pattern Analysis and Machine Intelligence
Clustering Algorithms
Density-Based Clustering in Spatial Databases: The Algorithm GDBSCAN and Its Applications
Data Mining and Knowledge Discovery
Discrete & Computational Geometry
Persistence barcodes for shapes
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Stability of Persistence Diagrams
Discrete & Computational Geometry
A tutorial on spectral clustering
Statistics and Computing
A Graph-Theoretic Approach to Nonparametric Cluster Analysis
IEEE Transactions on Computers
On the Local Behavior of Spaces of Natural Images
International Journal of Computer Vision
Introduction to Nonparametric Estimation
Introduction to Nonparametric Estimation
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Scalar Field Analysis over Point Cloud Data
Discrete & Computational Geometry
Least squares quantization in PCM
IEEE Transactions on Information Theory
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We present a clustering scheme that combines a mode-seeking phase with a cluster merging phase in the corresponding density map. While mode detection is done by a standard graph-based hill-climbing scheme, the novelty of our approach resides in its use of topological persistence to guide the merging of clusters. Our algorithm provides additional feedback in the form of a set of points in the plane, called a persistence diagram (PD), which provably reflects the prominences of the modes of the density. In practice, this feedback enables the user to choose relevant parameter values, so that under mild sampling conditions the algorithm will output the correct number of clusters, a notion that can be made formally sound within persistence theory. In addition, the output clusters have the property that their spatial locations are bound to the ones of the basins of attraction of the peaks of the density. The algorithm only requires rough estimates of the density at the data points, and knowledge of (approximate) pairwise distances between them. It is therefore applicable in any metric space. Meanwhile, its complexity remains practical: although the size of the input distance matrix may be up to quadratic in the number of data points, a careful implementation only uses a linear amount of memory and takes barely more time to run than to read through the input.