We show how to break the curse of dimension for the estimation of optimal transport distance between two smooth distributions for the Euclidean squared distance. The approach relies on essentially one tool: represent inequality constraints in the dual formulation of OT by equality constraints with a sum of squares in reproducing kernel Hilbert space. By showing this representation is tight in the variational formulation, one can then leverage smoothness to break the curse. (*) However, the constants associated with the algorithm a priori scale exponentially with the dimension.
In this talk, I present two very different applications related by the simple idea of invertible transformations.
- The first topic is an analysis of inverse consistency penalty in image matching in conjunction with the use of neural networks. We show that neural networks favours the emergence of smooth transformation for the inverse consistency. Experimentally, we show that this behaviour is fairly stable with respect to the chosen architecture. This is joint work with H. Greer, R. Kwitt and M. Niethammer.
-The second topic is an analysis of global convergence of residual networks when the residual block is parametrized via reproducing kernel Hilbert space vector field. We prove that the resulting problem satisfies the so-called Polyak-Lojasiewicz property, for instance ensuring global convergence if the iterates are bounded. We show that this property applies in a continuous limit as well as in the fully discrete setting. This is joint work with R. Barboni and G. Peyré.
I explain the formulation of the Wasserstein-Fisher-Rao distance, introduced a few years ago as the natural extension of the Wasserstein L2 metric to the space of positive Radon measures. We present the equivalence between the dynamic formulation and a static formulation, where we relax the marginal constraints with relative entropy. Then, we will present a connection with standard optimal transport on a cone space. The second part of the talk is motivated by this optimal transport distance and we study a generalized Camassa-Holm equation, for which we study existence of minimizing generalized geodesics, a la Brenier.