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Sparse neural networks are effective approaches to reduce the resource requirements for the deployment of deep neural networks. Recently, the concept of adaptive sparse connectivity, has emerged to allow training sparse neural networks from…
Numerical simulations of turbulent fluids are paramount to real-life applications, from predicting and modeling flows to diagnostic purposes in engineering. However, they are also computationally challenging due to their intrinsically…
We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in…
An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d \rightarrow \infty$ is Lipschitz and $\left\{x_1, \dots, x_N \right\} \subset [0,1]^d$, then $$ \left| \int_{[0,1]^d} f(x) dx - \frac{1}{N}…
A new method is developed for the study of transport properties of 1D models with random potentials. It is based on an exact transformation that reduces discrete Schr\"odinger equation in the tight-binding model to a two-dimensional…
Optimal transport (OT) is a popular measure to compare probability distributions. However, OT suffers a few drawbacks such as (i) a high complexity for computation, (ii) indefiniteness which limits its applicability to kernel machines. In…
Common measures of neural representational (dis)similarity are designed to be insensitive to rotations and reflections of the neural activation space. Motivated by the premise that the tuning of individual units may be important, there has…
Optimal transport induces the Earth Mover's (Wasserstein) distance between probability distributions, a geometric divergence that is relevant to a wide range of problems. Over the last decade, two relaxations of optimal transport have been…
An n-tuple (n \geq 2), T = (T_1, \ldots, T_n), of commuting bounded linear operators on a Hilbert space \mathcal{H} is doubly commuting if T_i T_j^* = T_j^* T_i for all $1 \leq i < j \leq n$. If in addition, each T_i \in C_{\cdot 0}, then…
For a measure preserving transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1<n,...,i_d<n}…
For a $d$-dimensional log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to a convex body $K$, the problem of outputting samples from a distribution $\nu$ which is $\varepsilon$-close in infinity-distance…
This work introduces novel computational methods for entropic optimal transport (OT) problems under martingale-type conditions. The considered problems include the discrete martingale optimal transport (MOT) problem. Moreover, as the…
We show non-asymptotic exponential convergence of Sinkhorn iterates to the Schr\"odinger potentials, solutions of the quadratic Entropic Optimal Transport problem on $\mathbb{R}^ d$. Our results hold under mild assumptions on the marginal…
The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality…
We combine a periodization strategy for weighted $L_{2}$-integrands with efficient approximation methods in order to approximate multivariate non-periodic functions on the high-dimensional cube $\left[-\frac{1}{2},\frac{1}{2}\right]^{d}$.…
We introduce sliced optimal transport dataset distance (s-OTDD), a model-agnostic, embedding-agnostic approach for dataset comparison that requires no training, is robust to variations in the number of classes, and can handle disjoint label…
This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and…
Let $D$ be the set of $n\times n$ positive semidefinite matrices of trace equal to one, also known as the set of density matrices. We prove two results on the hardness of approximating $D$ with polytopes. First, we show that if $0 <…
We prove consistency of a recently proposed scheme that evaluates expected values by composing a learned transport map with Clenshaw--Curtis sparse-grid quadrature on a tractable product source. Our analysis hinges on the structural fact…
We investigate the approximation of Monge--Kantorovich problems on general compact metric spaces, showing that optimal values, plans and maps can be effectively approximated via a fully discrete method. First we approximate optimal values…