Computational Mathematics and Scientific Computing Seminar

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Generative Modeling is the task of synthesizing new data given a training dataset. While there exist many techniques to solve this problem, diffusion models are now the de facto state-of-the-art method for generative modeling. They consist of a ‘noising’ stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a `denoising’ process defined by approximating the time-reversal of the diffusion. In this talk I discuss two extensions of the current paradigm of diffusion models. First, I will show how one can solve inverse problems and data translation problems leveraging those diffusion models. To do so I will introduce tools from Optimal Transport and in particular, the Schrödinger Bridge problem, i.e. an entropy-regularized optimal transport problem on path spaces. I will present Diffusion Schrödinger Bridge, an original approximation of the Iterative Proportional Fitting procedure to solve the Schrödinger Bridge problem. Then, I will turn to the extension of diffusion models to non Euclidean data. Indeed, classical generative models assume that data is supported on a Euclidean space. In many domains such as robotics, geoscience or protein modeling, data is often naturally described by distributions living on Riemannian manifolds which require new methodologies in order to be appropriately handled.