Related papers: Decomposition of geodesics in the Wasserstein spac…
In this paper, we study a connection between disintegration of measures and geometric properties of probability spaces. We prove a disintegration theorem, addressing disintegration from the perspective of an optimal transport problem. We…
Statistical inference based on optimal transport offers a different perspective from that of maximum likelihood, and has increasingly gained attention in recent years. In this paper, we study univariate nonparametric shape-constrained…
The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes synthetic notions of a Ricci curvature lower bound $K$ joint with an upper bound $N$ on the dimension. Their condition, called the Curvature-Dimension condition and…
We consider the space of probability measures on a discrete set $X$, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $Y \subseteq X$, it is natural to ask whether they can be connected…
Estimating a $d$-dimensional distribution $\mu$ by the empirical measure $\hat{\mu}_n$ of its samples is an important task in probability theory, statistics and machine learning. It is well known that $\mathbb{E}[\mathcal{W}_p(\hat{\mu}_n,…
We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete…
Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X)…
We review work on `decomposition,' a property of two-dimensional theories with 1-form symmetries and, more generally, d-dimensional theories with (d-1)-form symmetries. Decomposition is the observation that such quantum field theories are…
In this paper, we first show continuity of both Wasserstein projections in the convex order when they are unique. We also check that, in arbitrary dimension $d$, the quadratic Wasserstein projection of a probability measure $\mu$ on the set…
A set $V$ in a domain $U$ in $\mathbb{C}^n$ has the {\em norm-preserving extension property} if every bounded holomorphic function on $V$ has a holomorphic extension to $U$ with the same supremum norm. We prove that an algebraic subset of…
Consider an empirical measure $\mathbb{P}_n$ induced by $n$ iid samples from a $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$ and let $\gamma = N(0,\sigma^2 I_d)$ be the isotropic Gaussian measure. We study the speed of…
We study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Dualities for backward and forward projections are established under general conditions. Dual optimal solutions and their…
This paper studies the optimization of the KL functional on the Wasserstein space of probability measures, and develops a sampling framework based on Wasserstein gradient descent (WGD). We identify two important subclasses of the…
We show geodesic completeness of certain compact locally symmetric pseudo-Riemannian manifolds of signature $(2,n)$. Our model space $\mathbf{X}$ is a $1$-connected, indecomposable symmetric space of signature $(2,n)$, that admits a unique…
This paper proves an atomic decomposition of the space of $1$-dimensional metric currents without boundary, in which the atoms are specified by closed Lipschitz curves with uniform control on their Morrey norms. Our argument relies on a…
Generating samples given a specific label requires estimating conditional distributions. We derive a tractable upper bound of the Wasserstein distance between conditional distributions to lay the theoretical groundwork to learn conditional…
We study general geometric properties of cone spaces, and we apply them on the Hellinger--Kantorovich space $(\mathcal{M}(X),\mathsf{H\hspace{-0.25em} K}_{\alpha,\beta}).$ We exploit a two-parameter scaling property of the…
Fix probability densities $f$ and $g$ on open sets $X \subset \mathbf{R}^m$ and $Y \subset \mathbf{R}^n$ with $m\ge n\ge1$. Consider transporting $f$ onto $g$ so as to minimize the cost $-s(x,y)$. We give a non-degeneracy condition (a) on…
Non-negative matrix and tensor factorisations are a classical tool for finding low-dimensional representations of high-dimensional datasets. In applications such as imaging, datasets can be regarded as distributions supported on a space…
Despite the remarkable empirical success of generative models, the available theory on their statistical accuracy in scientific computing remains largely pessimistic. This paper develops a theoretical framework for understanding the…