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Register a new userBarycenters for the Hellinger--Kantorovich distance over $mathbb{R}^d$
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We study the barycenter of the Hellinger--Kantorovich metric over non-negative measures on compact, convex subsets of $mathbb{R}^d$. The article establishes existence, uniqueness (under suitable assumptions) and equivalence between a coupled-two-marginal and a multi-marginal formulation. We analyze the HK barycenter between Dirac measures in detail, and find that it differs substantially from the Wasserstein barycenter by exhibiting a local `clustering behaviour, depending on the length scale of the input measures. In applications it makes sense to simultaneously consider all choices of this scale, leading to a 1-parameter family of barycenters. We demonstrate the usefulness of this family by analyzing point clouds sampled from a mixture of Gaussians and inferring the number and location of the underlying Gaussians.
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In this paper we study the local linearization of the Hellinger--Kantorovich distance via its Riemannian structure. We give explicit expressions for the logarithmic and exponential map and identify a suitable notion of a Riemannian inner product. Samples can thus be represented as vectors in the tangent space of a suitable reference measure where the norm locally approximates the original metric. Working with the local linearization and the corresponding embeddings allows for the advantages of the Euclidean setting, such as faster computations and a plethora of data analysis tools, whilst still still enjoying approximately the descriptive power of the Hellinger--Kantorovich metric.
We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple 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, that 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.
We discuss a new notion of distance on the space of finite and nonnegative measures which can be seen as a generalization of the well-known Kantorovich-Wasserstein distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and its properties. In fact the distance can be equivalently described by an optimal transport problem on the cone space over the underlying metric space. We give a construction of geodesic curves and discuss their properties.
In this note we study and obtain factorization theorems for colorings of matrices and Grassmannians over $mathbb{R}$ and ${mathbb{C}}$, which can be considered metr
In this thesis, we consider the Wasserstein barycenter problem of discrete probability measures from computational and statistical sides in two scenarios: (I) the measures are given and we need to compute their Wasserstein barycenter, and (ii) the measures are generated from a probability distribution and we need to calculate the population barycenter of the distribution defined by the notion of Frechet mean. The statistical focus is estimating the sample size of measures necessary to calculate an approximation for Frechet mean (barycenter) of a probability distribution with a given precision. For empirical risk minimization approaches, the question of the regularization is also studied together with proposing a new regularization which contributes to the better complexity bounds in comparison with quadratic regularization. The computational focus is developing algorithms for calculating Wasserstein barycenters: both primal and dual algorithms which can be executed in a decentralized manner. The motivation for dual approaches is closed-forms for the dual formulation of entropy-regularized Wasserstein distances and their derivatives, whereas the primal formulation has closed-form expression only in some cases, e.g., for Gaussian measures. Moreover, the dual oracle returning the gradient of the dual representation for entropy-regularized Wasserstein distance can be computed for a cheaper price in comparison with the primal oracle returning the gradient of the entropy-regularized Wasserstein distance. The number of dual oracle calls, in this case, will also be less, i.e., the square root of the number of primal oracle calls. This explains the successful application of the first-order dual approaches for the Wasserstein barycenter problem.
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