相关论文: Quantum Earth mover's distance, No-go Quantum Kant…
Kantorovich distance (or 1-Wasserstein distance) on the probability simplex of a finite metric space is the value of a Linear Programming problem for which a closed-form expression is known in some cases. When the ground distance is defined…
A quantum version of the Monge--Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states $\rho^{AB}$, such that both of its reduced density matrices $\rho^A$ and…
An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators…
We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the optimal strategy that minimizes the total probability of error. This leads to the identification of the…
Given two distributions $P$ and $S$ of equal total mass, the Earth Mover's Distance measures the cost of transforming one distribution into the other, where the cost of moving a unit of mass is equal to the distance over which it is moved.…
Distinguishing quantum states with minimal sampling overhead is of fundamental importance to teach quantum data to an algorithm. Recently, the quantum Wasserstein distance emerged from the theory of quantum optimal transport as a promising…
The Earth movers distance (EMD) is a measure of distance between probability distributions which is at the heart of mass transportation theory. Recent research has shown that the EMD plays a crucial role in studying the potential impact of…
A novel neural architecture was recently developed that enforces an exact upper bound on the Lipschitz constant of the model by constraining the norm of its weights in a minimal way, resulting in higher expressiveness compared to other…
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}…
The goal of this thesis is to study the use of the Kantorovich-Rubinstein distance as to build a descriptor of sample complexity in classification problems. The idea is to use the fact that the Kantorovich-Rubinstein distance is a metric in…
The classical Kantorovich-Rubinstein duality guarantees coincidence between metrics on the space of probability distributions defined on the one hand via transport plans (couplings) and on the other hand via price functions. Both…
A representation for the Kantorovich--Rubinstein distance between probability measures on an abstract Wiener space in terms of the extended stochastic integral (or, divergence) operator is obtained.
Structured equations are a standard modeling tool in mathematical biology. They areintegro-differential equations where the unknown depends on one or several variables, representing the state or phenotype of individuals. A large literature…
The classical Monge-Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by…
Optimal transport theory has recently been extended to quantum settings, where the density matrices generalize the probability measures. In this paper, we study the computational aspects of the order 2 quantum Wasserstein distance,…
The Wasserstein distances $W_p$ ($p\geq 1$), defined in terms of solution to the Monge-Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou-Brenier formula characterizes the…
An upper bound for the Kantorovich transport distance between probability measures on multidimensional Euclidean spaces is given in terms of transport distances between one dimensional projections. This quantifies the Cram\'er-Wold…
The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space…
The Kantorovich distance is a widely used metric between probability distributions. The Kantorovich-Rubinstein duality states that it can be defined in two equivalent ways: as a supremum, based on non-expansive functions into [0, 1], and as…
This article presents a new class of distances between arbitrary nonnegative Radon measures inspired by optimal transport. These distances are defined by two equivalent alternative formulations: (i) a dynamic formulation defining the…