Witnessed k-Distance

Distance function to a compact set plays a central role in several areas of computational geometry. Methods that rely on it are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The recently introduced distance to a measure offers a solution by extending the distance function framework to reasoning about the geometry of probability measures, while maintaining theoretical guarantees about the quality of the inferred information. A combinatorial explosion hinders working with distance to a measure as an ordinary power distance function. In this paper, we analyze an approximation scheme that keeps the representation linear in the size of the input, while maintaining the guarantees on the inference quality close to those for the exact but costly representation.

[1]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[2]  Franz Aurenhammer,et al.  A new duality result concerning voronoi diagrams , 1986, ICALP.

[3]  Herbert Edelsbrunner,et al.  The union of balls and its dual shape , 1993, SCG '93.

[4]  H. Edelsbrunner.  The union of balls and its dual shape , 1995 .

[5]  Marshall W. Bern,et al.  Surface Reconstruction by Voronoi Filtering , 1998, SCG '98.

[6]  Sanjoy Dasgupta,et al.  Learning mixtures of Gaussians , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[7]  P. Massart,et al.  Adaptive estimation of a quadratic functional by model selection , 2000 .

[8]  C. Villani.  Topics in Optimal Transportation , 2003 .

[9]  Alexander Russell,et al.  Computational topology: ambient isotopic approximation of 2-manifolds , 2003, Theor. Comput. Sci..

[10]  Tamal K. Dey,et al.  Provable surface reconstruction from noisy samples , 2004, SCG '04.

[11]  Leonidas J. Guibas,et al.  The Earth Mover's Distance as a Metric for Image Retrieval , 2000, International Journal of Computer Vision.

[12]  Piotr Indyk,et al.  Nearest Neighbors in High-Dimensional Spaces , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[13]  C. Villani,et al.  Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces , 2005, math/0503123.

[14]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2005, Discret. Comput. Geom..

[15]  Trevor Darrell,et al.  Nearest-Neighbor Searching and Metric Space Dimensions , 2006 .

[16]  Frédéric Chazal,et al.  A Sampling Theory for Compact Sets in Euclidean Space , 2006, SCG '06.

[17]  David Cohen-Steiner,et al.  Vines and vineyards by updating persistence in linear time , 2006, SCG '06.

[18]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[19]  Steve Oudot,et al.  Towards persistence-based reconstruction in euclidean spaces , 2007, SCG '08.

[20]  Afra Zomorodian,et al.  Computational topology , 2010 .

[21]  Frédéric Chazal,et al.  Geometric Inference for Measures based on Distance Functions , 2011 .

[22]  Stephen Smale,et al.  A Topological View of Unsupervised Learning from Noisy Data , 2011, SIAM J. Comput..

[23]  Frédéric Chazal,et al.  Geometric Inference for Probability Measures , 2011, Found. Comput. Math..

[24]  Benoît R. Kloeckner.  Approximation by finitely supported measures , 2010, 1003.1035.

[25]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[26]  H. Edelsbrunner,et al.  Persistent Homology — a Survey , 2022 .