Project A2: Wasserstein metrics and their applications

Wasserstein metrics are nowadays a widespread tool in data science for processing objects, such as images, that can be interpreted as finite measures. In the last few years many new approaches to derive efficient algorithms for the computation of Wasserstein distances have been introduced.

Our first goal is to provide efficient algorithms to find barycenters in an incomplete Wasserstein metric for sets of point patterns in Rd, where incomplete refers to the fact that not the total mass has to be transported but untransported mass incurs additional cost. The results are applied to spatio-temporal data analysis. Further, we want to connect new ideas for Wasserstein dictionary learning with our own recent approaches in Project A6 for multi-scale dictionary learning based on graph structures. We are especially interested in structured dictionaries for analysis, denoising and sparse representation of special classes of images or tensors, particularly seismic images/tensors and fingerprint images.

Methods: optimal transport, k-means, multi-scale methods, non-linear approximation
Applications: spatio-temporal data analysis, fingerprint analysis, image similarity measures