Persistent Homology meets Statistical Inference – A Case Study: Detecting Modes of One-Dimensional Signals ∗ Ulrich Bauer 1 , Axel Munk 2,3 , Hannes Sieling 2 , and Max Wardetzky 4 1 IST Austria 2 Institute for Mathematical Stochastics, University of Göttingen 3 Max Planck Institute for Biophysical Chemistry, Göttingen 4 Institute of Numerical and Applied Mathematics, University of Göttingen January 12, 2015 Abstract We investigate the problem of estimating persistent homology of noisy one dimensional signals. We relate this to 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 extend the ideas of persistent homology by working with norms different from the (classical) supremum norm. As a particular case we investigate the so called Kolmogorov norm. We argue that this extension 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 so-called taut strings minimize the number of critical points for a very general class of functions. We illustrate our results by several numerical examples. AMS subject classification: Primary 62G05,62G20; secondary 62H12 1 Introduction Persistent homology and topological data analysis (TDA) Persistent homology [20], a rel- atively novel branch of algebraic topology, provides a quantitative notion of the stability or robustness of critical points of a (sufficiently nice) real valued function f on a topological space: the persistence of a critical point is a lower bound on the amount of perturbation (in the supremum norm) required for its elimination. Persistence is measured by the life span of a critical point – the difference between its birth and death – according to some filtration of the underlying topological space. Filtrations arise, for example, by considering sublevel sets of f . Birth and ∗ Research partially supported by DFG FOR 916, Volkswagen Foundation, and the Toposys project FP7-ICT- 318493-STREP 1 arXiv:1404.1214v1 [math.ST] 4 Apr 2014