Abstract: We show that the geometric and the metric approach to the definition of Sobolev spaces on spaces of (probability) measures coincide, provided suitable reference measures are chosen. For a specific (and in some way, canonical) choice of such a measure, we consider the induced Dirichlet form, which is naturally associated to a diffusion process on the space of non-negative measures endowed with the Hellinger–Kantorovich distance. Applications to image classification are also presented. This work has been done in collaboration with Lorenzo Dello Schiavo, Massimo Fornasier, and Pascal Heid.

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