### Persistence-Based Clustering in Riemannian ManifoldsPersistence-Based Clustering in Riemannian Manifolds

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 Author Chazal, Frdric ♦ Guibas, Leonidas J. ♦ Oudot, Steve Y. ♦ Skraba, Primoz Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2013 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Morse theory ♦ Unsupervised learning ♦ Clustering ♦ Computational topology ♦ Mode seeking ♦ Topological persistence Abstract 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 $\textit{correct}$ 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. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2013-11-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 60 Issue Number 6 Page Count 38 Starting Page 1 Ending Page 38

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Source: ACM Digital Library