Geometric estimation of probability measures in high-dimensions

We are interested in constructing adaptive probability models for high-dimensional data that is well-approximated by low-dimensional geometric structures. We discuss a family of estimators for probability distributions based on data-adaptive multiscale geometric approximations. They are particularly effective when the probability distribution concentrates near low-dimensional sets, having sample and computational complexity depending mildly (linearly in cases of interest) in the ambient dimension, as well as in the intrinsic dimension of the data, suitably defined. Moreover the construction of these estimators may be performed, under suitable assumptions, by fast algorithms, with cost O((c^d ; d²)Dnlog n) where n is the number of samples, D the ambient dimension, d is the intrinsic dimension of the data, and c a small constant.