  Data analysis in high-dimensional spaces aims at obtaining a synthetic
description of a data set, revealing its main structure and its salient
features. We here introduce an approach providing this description in the form
of a topography of the data, namely a human-readable chart of the probability
density from which the data are harvested. The approach is based on an
unsupervised extension of Density Peak clustering and a non-parametric density
estimator that measures the probability density in the manifold containing the
data. This allows finding automatically the number and the height of the peaks
of the probability density, and the depth of the "valleys" separating them.
Importantly, the density estimator provides a measure of the error, which
allows distinguishing genuine density peaks from density fluctuations due to
finite sampling. The approach thus provides robust and visual information about
the density peaks' height, their statistical reliability, and their
hierarchical organization, offering a conceptually powerful extension of the
standard clustering partitions. We show that this framework is particularly
useful in the analysis of complex data sets.
