Gabriele Todeschi (Université Gustave Eiffel): L1 optimal transport for vector valued measures and application to Full Waveform Inversion

There is an increasing interest nowadays into applying optimal transport to define loss functions in inverse problems. However, optimal transport is intrinsically defined for non-negative (probability) measures and generalizations of the model and/or manipulations of the data are necessary in order to deal with general data space, frequently signed functions. A possible way to recover positivity is to lift the data in a higher dimensional vector space. For example, signed multi-component signals can be naturally lifted into positive semi-definite matrices. We introduce for this reason an unbalanced formulation of the L^1 optimal transport problem for vector valued measures. The favorable computational complexity of the L^1 problem, an advantage compared to other formulations of optimal transport which is fundamental for applications to huge data set, is inherited by our vector extension. We consider both a one-homogeneous and a two-homogeneous penalization for the imbalance of mass, the latter being potentially relevant for applications to physics based problems. In particular, we demonstrate the potential of our strategy for Full Waveform Inversion, an inverse problem for high resolution seismic imaging.