Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures

By Matthias Liero, Alexander Mielke, Giuseppe Savar'e. 2015

406 citations recorded by the source corpus.

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger–Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger–Kakutani and Kantorovich–Wasserstein distances.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/7eed503702a760d49c744ada559481ce6cfd6502

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In Applied Mathematics

Citations in the corpus

Scaling Algorithms for Unbalanced Transport Problems (2016, background): “The following result is a minor extension of results from [47, 24] and is useful if one wishes to vary the parameter λ in numerical applications.”
A new optimal transport distance on the space of finite Radon measures (2015, methodology): “After finalization of this article we became aware of the parallel and completely independent works [16, 34], whose preprints appeared almost simultaneously with ours.”
Unbalanced optimal transport: Dynamic and Kantorovich formulations (2015, background): “The construction of [16] deﬁnes three equivalent static formulations.”
Wasserstein Auto-Encoders (2017, neutral): “the objective with λ · ( Df (ΓX , P ) +Df (ΓY , Q) ) [15, 16], where ΓX and ΓY are marginals of Γ.”
Computational Optimal Transport (2018, background): “In the special case c(x, y) = ‖x− y‖2, Dφ = KL, Lc (α, β)1/2 is the Gaussian-Hellinger distance [Liero et al., 2015], and it is shown to be a distance onM(1)+(R).”
Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation (2020, methodology): “[6, 13] have proposed one such relaxation using f -divergence on marginal distributions.”
{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, background): “This very interesting distance has been independently introduced by three different groups around the year 2015, and we refer to [31,32,60,63,64] for the different presentations and the different considerations which have been investigated so far.”
TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics (2020, background): “Another direction that has recently received increased attention is in unbalanced optimal transport where the goal is to relax the problem to add and remove mass (Benamou, 2003; Chizat et al., 2018; Liero et al., 2018; Schiebinger et al., 2019).”
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties (2018, neutral): “(P2(X), dW ), where X is an RCD(K,∞) metric-measure space and φ is the relative entropy functional [88, 69], and there is an intensive research to construct EVI gradient flows in ad-hoc geometric settings for reaction-diffusion equations [54, 25, 57, 55] and systems [65, 48, 56], nonlinear viscoelas”