An unsupervised procedure for reconstructing the topography of complex probability distributions.

We introduce an accurate and parameter free approach to estimate the probability density for non-uniform samples embedded in high dimensional and curved manifolds.
The estimator uses only distances between the points and not their coordinates and does not require specifying the functional form of the probability. Moreover, it provides an approximately unbiased measure of the uncertainty of the estimate.
The availability of an error estimate allows distinguishing genuine probability peaks from statistical fluctuations due to finite sampling. This is an essential ingredient for  finding the meaningful  peaks and the saddles between them.  Based on this information, we then reconstruct a hierarchical topography of the probability density, which provides a compact representation of the data set.
We demonstrate the usefulness of the approach in classification problems in functional magnetic resonance imaging, proteomics and metagenomics.