Persistent Homology meets Statistical Inference - A Case Study: Detecting Modes of One-Dimensional Signals

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.