Many datasets in modern applications - from cell gene expression and images to shapes and text documents - are naturally interpreted as probability measures, distributions, histograms, or point clouds. This perspective motivates the development of learning algorithms that operate directly in the space of probability measures. However, this space presents unique challenges: it is nonlinear and infinite-dimensional. Fortunately, it possesses a natural Riemannian-type geometry which enables meaningful learning algorithms.

This talk will provide an introduction to the space of probability measures and present approaches to unsupervised, supervised, and manifold learning within this framework. We will examine temporal evolutions on this space, including flows involving stochastic gradient descent and trajectory inference, with applications to analyzing gene expression in single cells. The proposed algorithms are furthermore demonstrated in pattern recognition tasks in imaging and medical applications.