Beyond Wasserstein geodesic: spline interpolation for distributions
Recently optimal mass transport (OMT) theory has attracted a lot of attention due to its effectiveness in a range of applications in particular machine learning. Among other things, OMT induces a natural interpolation between two probability distributions. This Wasserstein geodesic, or more precisely displacement interpolation, between two distributions is a counterpart of linear interpolation in the space of probability distributions. This has been used to interpolate images, histograms etc. A most natural extension of Wasserstein geodesic is a smooth interpolation of multiple distribution. In this talk, I will discuss measure-valued spline, which generalizes the notion of cubic spline to the space of distributions. It provides one elegant solution to the problem to smoothly interpolate (empirical) probability measures. Potential applications include time sequence interpolation or regression of images, histograms or aggregated datas. I will focus on theory but algorithms will also be briefly mentioned.