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Abstract:
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The space of probability measures P(X) on a metric space X is known to inherit a metric structure through the Optimal Transport distance, a.k.a Wasserstein distance, which computes the minimal cost to move mass from one measure to another. When the space X is moreover a Riemannian manifold, or is just the Euclidean space, the Wasserstein space also inherits this Riemannian structure, where concepts such as geodesics or tangent vectors can be rigorously defined. This Riemannian structure is the key to adapt some classic Machine Learning algorithms, usually well known on Euclidean / Hilbert spaces, to the Wasserstein space. We will show for example how to parameterize and optimize problems such as Principal Component Analysis and Dictionary Learning, when the data set is made of histograms or discrete probability measures.
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