The GenCol algorithm for high-dimensional optimal transport: general formulation and application to barycenters and Wasserstein splines

We extend the recently introduced genetic column generation algorithm for high-dimensional multi-marginal optimal transport from symmetric to general problems. We use the algorithm to calculate accurate mesh-free Wasserstein barycenters and cubic Wasserstein splines

Convex geometry of finite exchangeable laws and de Finetti style representation with universal correlated corrections

We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If $(Z_1,...,Z_N)$ is a finitely exchangeable sequence of $N$ random variables taking values in some Polish space $X$, we show that the law $\mu_k$ of the first $k$ components has a representation of the form \mu_k=\int_{{\mathcal P}_{\frac{1}{N}}(X)} F_{N,k}(\lambda) \, \mbox{d} \alpha(\lambda) for some probability measure $\alpha$ on the set of $1/N$-quantized probability measures on $X$ and certain universal polynomials $F_{N,k}$. The latter consist of a leading term $N^{k-1}\! /{\small \prod_{j=1}^{k-1}(N\! -\! j)\, \lambda^{\otimes k}}$ and a finite, exponentially decaying series of correlated corrections of order $N^{-j}$ ($j=1,...,k$). The $F_{N,k}(\lambda)$ are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals $\lambda$. Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis-Freedman between finite and infinite exchangeable laws

The strong interaction limit of DFT: what's known, what's new, what's open (REVIEW)

I will survey main results (both at the rigorous and the nonrigorous level) and open questions on the strongly correlated limit of DFT, including: - the connection between Hohenberg-Kohn-Lieb-Levy constrained-search and minimization of the interaction energy over $|\Psi|^2$ (alias Kantorovich optimal transport) - the SCE (alias Monge) ansatz in the Kantorovich problem: where it works, where it fails - the new quasi-Monge ansatz [1] which - unlike the SCE ansatz - always yields the minimum Kantorovich cost, but whose data complexity scales linearly instead of exponentially with the number of particles/marginals - asymptotic and semi-empirical exchange-correlation functionals related to the strictly correlated limit - representability challenges. [1] G.Friesecke, D.Vögler, Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces, SIAM J. Math. Analysis Vol. 50 No. 4, 3996-4019, 2018Non UBCUnreviewedAuthor affiliation: Technische Universitat MunichFacult