Analyzing large volumes of high-dimensional data is an issue of fundamental importance in science
and beyond. Several approaches work on the assumption that the important content of a dataset
belongs to a manifold whose Intrinsic Dimension (ID) is much lower than the crude large number
of coordinates. That manifold however is generally twisted and curved; in addition points on it will
be non-uniformly distributed: two factors that make the identification of the ID and its exploitation
really hard. Here we propose a new ID estimator using only the distance of the first and the second
nearest neighbor of each point in the sample. This extreme minimality enables us to reduce the
effects of curvature, of density variation, and the resulting computational cost. The ID estimator is
theoretically exact in uniformly distributed data sets, and provides consistent measures in general.
When used in combination with block analysis, it allows discriminating the relevant dimensions as
a function of the block size. This allows estimating the ID even when the data lie on a manifold
perturbed by a high-dimensional noise, a situation often encountered in real world data sets. Upon defining a notion of distance between protein sequences, This tools is used to estimate the ID of protein families, and to assess the consistency of generative models. Moreover, If coupled with a density estimator, our ID allows to measure the density of points by taking into account the space in which they actually lie, thus allowing for a cleaner estimation. Here we move a step further towards an automatic classification of protein sequences by using three new tools: our ID estimator, a density estimator and a clustering algorithm. We present the analysis performed on a Pfam PUA clan, showing that these combined tools allow to successfully separate protein domains into architectures. Finally, we present a generalized model for the estimation of the ID that is able to work in data sets with multiple dimensionalities: taking advantage of Bayesian inference techniques, the method allows discriminating manifolds with different dimensions as well as assigning all the points to the respective manifolds. We test the method on a molecular dynamics trajectory, showing that the folded state has a higher dimension with respect to the unfolded one
