Related papers: Lifting couplings in Wasserstein spaces
We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $\R^n$ in a way that is completely algebraic.…
In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. The arrow categories are more simple forms of the \emph{comma} categories and were introduced…
We develop flexible methods of deriving variational inference for models with complex latent variable structure. By splitting the variables in these models into "global" parameters and "local" latent variables, we define a class of…
At least two, different approaches to define and solve statistical models for the analysis of economic systems exist: the typical, econometric one, interpreting the Gravity Model specification as the expected link weight of an arbitrary…
Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order theory on a given finite domain. WFOMC has emerged as a fundamental tool for probabilistic inference. Algorithms for WFOMC that run in…
It has been shown beneficial for many types of data which present an underlying hierarchical structure to be embedded in hyperbolic spaces. Consequently, many tools of machine learning were extended to such spaces, but only few…
In this paper, we present a novel approach for analyzing the relationship between the supports of conditional measures and their geometric arrangement in Wasserstein space via the disintegration map. Our method establishes criteria to…
Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group…
We present an algebraic account of the Wasserstein distances $W_p$ on complete metric spaces, for $p \geq 1$. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms,…
We are interested in martingale rearrangement couplings. As introduced by Wiesel [37] in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted Wasserstein distance of couplings between two…
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…
A variety of lifted inference algorithms, which exploit model symmetry to reduce computational cost, have been proposed to render inference tractable in probabilistic relational models. Most existing lifted inference algorithms operate only…
In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space $(\mathcal{P}_2(H),W_2)$ of Borel probability measures with finite quadratic moment on a separable Hilbert space $H$. We will show that…
We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a…
We prove general upper estimates for the distance between two Borel probability measures in Wasserstein metric in terms of the Fourier transforms of the measures. We work in compact manifolds including the torus, the Euclidean unit sphere,…
We provide a geometric interpretation to Bayesian inference that allows us to introduce a natural measure of the level of agreement between priors, likelihoods, and posteriors. The starting point for the construction of our geometry is the…
We initiate a study of varieties of minimal degree in weighted projective spaces. We call a weighted projective space $\mathbf{P}(w_0,\dots,w_n)$ divisible if $w_i \mid w_{i+1}$ for all $i$. We provide sharp bounds for when a non-degenerate…
We present necessary and sufficient conditions for a group homomorphism between spaces of smooth sections of Lie group bundles to be a weighted composition operator. These results provide new insights into a wide range of problems related…
We formulate and solve a regression problem with time-stamped distributional data. Distributions are considered as points in the Wasserstein space of probability measures, metrized by the 2-Wasserstein metric, and may represent images,…
Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics…