Persistence Barcodes Versus Kolmogorov Signatures: Detecting Modes of One-Dimensional Signals

We investigate the problem of estimating the number of modes (i.e., local maxima)—a well-known question in statistical inference—and we show how to do so without presmoothing the data. To this end, we modify the ideas of persistence barcodes by first relating persistence values in dimension one to distances (with respect to the supremum norm) to the sets of functions with a given number of modes, and subsequently working with norms different from the supremum norm. As a particular case, we investigate the Kolmogorov norm. We argue that this modification has certain statistical advantages. We offer confidence bands for the attendant Kolmogorov signatures, thereby allowing for the selection of relevant signatures with a statistically controllable error. As a result of independent interest, we show that taut strings minimize the number of critical points for a very general class of functions. We illustrate our results by several numerical examples.

[1]  Markus Grasmair,et al.  The Equivalence of the Taut String Algorithm and BV-Regularization , 2006, Journal of Mathematical Imaging and Vision.

[2]  I. Good,et al.  Density Estimation and Bump-Hunting by the Penalized Likelihood Method Exemplified by Scattering and Meteorite Data , 1980 .

[3]  Sivaraman Balakrishnan,et al.  Minimax rates for homology inference , 2011, AISTATS.

[4]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

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

[6]  Yu. I. Ingster,et al.  Nonparametric Goodness-of-Fit Testing Under Gaussian Models , 2002 .

[7]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[8]  B. Silverman,et al.  Using Kernel Density Estimates to Investigate Multimodality , 1981 .

[9]  R. Adler,et al.  Crackle: The Persistent Homology of Noise , 2013, 1301.1466.

[10]  J. Hartigan.  Testing for Antimodes , 2000 .

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

[12]  H. Chan,et al.  Detection with the scan and the average likelihood ratio , 2011, 1107.4344.

[13]  Sivaraman Balakrishnan,et al.  Confidence sets for persistence diagrams , 2013, The Annals of Statistics.

[14]  D. Donoho.  One-sided inference about functionals of a density , 1988 .

[15]  Matthew P. Wand,et al.  Kernel Smoothing , 1995 .

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

[17]  S. Geer,et al.  Locally adaptive regression splines , 1997 .

[18]  D. Donoho,et al.  Higher criticism for detecting sparse heterogeneous mixtures , 2004, math/0410072.

[19]  I. Johnstone,et al.  Wavelet Shrinkage: Asymptopia? , 1995 .

[20]  Herbert Edelsbrunner,et al.  Topological persistence and simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[21]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[22]  E. Rio,et al.  Théorie asymptotique de processus aléatoires faiblement dépendants , 2000 .

[23]  Peter Bubenik,et al.  A statistical approach to persistent homology , 2006, math/0607634.

[24]  P. Davies,et al.  Densities, spectral densities and modality , 2004, math/0410071.

[25]  Jean-Paul Chilès,et al.  Wiley Series in Probability and Statistics , 2012 .

[26]  Leonidas J. Guibas,et al.  Gromov‐Hausdorff Stable Signatures for Shapes using Persistence , 2009, Comput. Graph. Forum.

[27]  J. Harer,et al.  Improving homology estimates with random walks , 2011 .

[28]  Ulrich Bauer,et al.  Optimal Topological Simplification of Discrete Functions on Surfaces , 2012, Discret. Comput. Geom..

[29]  J. Hartigan,et al.  The Dip Test of Unimodality , 1985 .

[30]  Matthew Kahle,et al.  Random Geometric Complexes , 2009, Discret. Comput. Geom..

[31]  M. Grasmair,et al.  Generalizations of the Taut String Method , 2008 .

[32]  Gunnar Carlsson,et al.  Topological De-Noising: Strengthening the Topological Signal , 2009, ArXiv.

[33]  Sivaraman Balakrishnan,et al.  Statistical Inference For Persistent Homology , 2013, arXiv.org.

[34]  Jon A. Wellner,et al.  Empirical Processes with Applications to Statistics. , 1988 .

[35]  G. Carlsson,et al.  Statistical topology via Morse theory, persistence and nonparametric estimation , 2009, 0908.3668.

[36]  P. Davies,et al.  Local Extremes, Runs, Strings and Multiresolution , 2001 .

[37]  Don Sheehy,et al.  A Multicover Nerve for Geometric Inference , 2012, CCCG.

[38]  Maurizio Vichi,et al.  Studies in Classification Data Analysis and knowledge Organization , 2011 .