  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.
