From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data clouds
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Publication:6339063
DOI10.1007/S00205-022-01770-8arXiv2004.09304MaRDI QIDQ6339063
Ryan W. Murray, Nicolás García Trillos, Matthew Thorpe
Publication date: 20 April 2020
Abstract: In this work we study statistical properties of graph-based clustering algorithms that rely on the optimization of balanced graph cuts, the main example being the optimization of Cheeger cuts. We consider proximity graphs built from data sampled from an underlying distribution supported on a generic smooth compact manifold . In this setting, we obtain high probability convergence rates for both the Cheeger constant and the associated Cheeger cuts towards their continuum counterparts. The key technical tools are careful estimates of interpolation operators which lift empirical Cheeger cuts to the continuum, as well as continuum stability estimates for isoperimetric problems. To our knowledge the quantitative estimates obtained here are the first of their kind.
Nonparametric estimation (62G05) Graph algorithms (graph-theoretic aspects) (05C85) Theory of data (68Pxx) General convexity (52Axx) Manifolds and measure-geometric topics (49Qxx)
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