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Eulerian methods for inverse problems using optimal transport

  • Heitz, Matthieu
Publication Date
May 18, 2020
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The goal of this thesis is to develop new numerical methods to address inverse problems using optimal transport. Inverse problems appear in many disciplines such as astronomy, geophysics or medical imaging, but also in fields closer to the focus of this thesis, namely computer vision, computer graphics, and machine learning. They are difficult problems by nature as they are often not well posed (infinite number of solutions, instabilities), and those that involve non-linear models such as optimal transport yield additional challenges. However, they are important problems to solve since they give access to quantities that are not directly observable, which provides major insight in many cases. Existing techniques for inverse problems in signal/image processing and machine learning often treat histograms as Euclidean data, hence they fail to grasp the underlying relationships between the bins, defined by the geometry of the domain. Optimal transport addresses this issue by building on the distance between bins to produce a distance between histograms (and more generally probability distributions). In this thesis, we adapt two well-known machine learning tasks to the optimal transport framework: dictionary learning and metric learning. Our methods address these tasks as optimization problems and rely on the entropic regularization of optimal transport, and automatic differentiation. The regularization provides fast, robust and smooth approximations of the transport, which are essential features for efficient optimization schemes. Automatic differentiation provides a fast and reliable alternative to manual analytical derivation, resulting in flexible frameworks. We motivate our algorithms with applications in image processing and natural language processing

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