Statistical Inference using the Morse-Smale Complex

The Morse-Smale complex of a function $f$ decomposes the sample space into cells where $f$ is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize, and compare multivariate functions. In this paper, we present some statistical results on estimating Morse-Smale complexes. This allows us to derive new results for two existing methods: mode clustering and Morse-Smale regression. We also develop two new methods based on the Morse-Smale complex: a visualization technique for multivariate functions and a two-sample, multivariate hypothesis test.

[1]  M. Morse.  Relations between the critical points of a real function of $n$ independent variables , 1925 .

[2]  Marston Morse.  The Foundations of a Theory of the Calculus of Variations in the Large in m-Space (Second Paper) , 1930 .

[3]  E. Nadaraya.  On Estimating Regression , 1964 .

[4]  J. Kruskal.  Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis , 1964 .

[5]  William M. Rand,et al.  Objective Criteria for the Evaluation of Clustering Methods , 1971 .

[6]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[7]  H. D. Brunk,et al.  Statistical inference under order restrictions : the theory and application of isotonic regression , 1973 .

[8]  Larry D. Hostetler,et al.  The estimation of the gradient of a density function, with applications in pattern recognition , 1975, IEEE Trans. Inf. Theory.

[9]  S. Helgason.  Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

[10]  L. Hubert,et al.  Comparing partitions , 1985 .

[11]  C. D. Kemp,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[12]  Peter Bacchetti,et al.  Additive Isotonic Models , 1989 .

[13]  G. E. Bredon.  Topology and geometry , 1993 .

[14]  Yizong Cheng,et al.  Mean Shift, Mode Seeking, and Clustering , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  R. Bhatia.  Matrix Analysis , 1996 .

[16]  Dorin Comaniciu,et al.  Mean Shift: A Robust Approach Toward Feature Space Analysis , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  L. Baringhaus,et al.  On a new multivariate two-sample test , 2004 .

[18]  G. Székely,et al.  TESTING FOR EQUAL DISTRIBUTIONS IN HIGH DIMENSION , 2004 .

[19]  L. Wasserman.  All of Nonparametric Statistics , 2005 .

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

[21]  Maria L. Rizzo,et al.  A new test for multivariate normality , 2005 .

[22]  Gábor J. Székely,et al.  Hierarchical Clustering via Joint Between-Within Distances: Extending Ward's Minimum Variance Method , 2005, J. Classif..

[23]  Uwe Einmahl,et al.  Uniform in bandwidth consistency of kernel-type function estimators , 2005 .

[24]  Surajit Ray,et al.  A Nonparametric Statistical Approach to Clustering via Mode Identification , 2007, J. Mach. Learn. Res..

[25]  Nicola Torelli,et al.  Clustering via nonparametric density estimation , 2007, Stat. Comput..

[26]  Tarn Duong,et al.  ks: Kernel Density Estimation and Kernel Discriminant Analysis for Multivariate Data in R , 2007 .

[27]  Frédo Durand,et al.  A Topological Approach to Hierarchical Segmentation using Mean Shift , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[28]  A. Tsybakov,et al.  Fast learning rates for plug-in classifiers , 2007, 0708.2321.

[29]  R. Brinkman,et al.  High-content flow cytometry and temporal data analysis for defining a cellular signature of graft-versus-host disease. , 2007, Biology of blood and marrow transplantation : journal of the American Society for Blood and Marrow Transplantation.

[30]  David Hinkley,et al.  Bootstrap Methods: Another Look at the Jackknife , 2008 .

[31]  Robert D. Nowak,et al.  Adaptive Hausdorff Estimation of Density Level Sets , 2009, COLT.

[32]  Stefano Soatto,et al.  Quick Shift and Kernel Methods for Mode Seeking , 2008, ECCV.

[33]  James Bailey,et al.  Information theoretic measures for clusterings comparison: is a correction for chance necessary? , 2009, ICML '09.

[34]  A. Rinaldo,et al.  Generalized density clustering , 2009, 0907.3454.

[35]  Maria L. Rizzo,et al.  DISCO analysis: A nonparametric extension of analysis of variance , 2010, 1011.2288.

[36]  Valerio Pascucci,et al.  Visual Exploration of High Dimensional Scalar Functions , 2010, IEEE Transactions on Visualization and Computer Graphics.

[37]  L. Wasserman,et al.  The Geometry of Nonparametric Filament Estimation , 2010, 1003.5536.

[38]  M. Wand,et al.  ASYMPTOTICS FOR GENERAL MULTIVARIATE KERNEL DENSITY DERIVATIVE ESTIMATORS , 2011 .

[39]  Samuel Gerber,et al.  Data Analysis with the Morse-Smale Complex: The msr Package for R , 2012 .

[40]  T. Duong,et al.  Data-driven density derivative estimation, with applications to nonparametric clustering and bump hunting , 2012, 1204.6160.

[41]  Larry A. Wasserman,et al.  Nonparametric Ridge Estimation , 2012, ArXiv.

[42]  Rebecca Nugent,et al.  Stability of density-based clustering , 2010, J. Mach. Learn. Res..

[43]  Oliver Rübel,et al.  Morse–Smale Regression , 2013, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[44]  Tarn Duong,et al.  Local significant differences from nonparametric two-sample tests , 2013 .

[45]  Maria L. Rizzo,et al.  Energy statistics: A class of statistics based on distances , 2013 .

[46]  L. Wasserman,et al.  A Comprehensive Approach to Mode Clustering , 2014, 1406.1780.

[47]  José E. Chacón,et al.  A Population Background for Nonparametric Density-Based Clustering , 2014, 1408.1381.

[48]  Larry A. Wasserman,et al.  Risk Bounds For Mode Clustering , 2015, ArXiv.

[49]  Christopher R. Genovese,et al.  Asymptotic theory for density ridges , 2014, 1406.5663.

[50]  David Mason,et al.  On the Estimation of the Gradient Lines of a Density and the Consistency of the Mean-Shift Algorithm , 2016, J. Mach. Learn. Res..

[51]  Frédéric Chazal,et al.  Robust Topological Inference: Distance To a Measure and Kernel Distance , 2014, J. Mach. Learn. Res..