This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to …
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)H\older continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent H\older-stability results for optimal transport maps.
We study the quantitative stability of the quadratic optimal transport map between a fixed probability density ρ and a probability measure μ on Rd, which we denote Tμ. Assuming that …
We study the quantitative stability of the quadratic optimal transport map between a fixed probability density ρ and a probability measure μ on Rd, which we denote Tμ. Assuming that the source density ρ is bounded from above and below on a compact convex set, we prove that the map μ↦Tμ is bi-Hölder continuous on large families of probability measures, such as the set of probability measures whose moment of order p>d is bounded by some constant. These stability estimates show that the linearized optimal transport metric W2,ρ(μ,ν)=‖Tμ−Tν‖L2(ρ,Rd) is bi-Hölder equivalent to the 2-Wasserstein distance on such sets, justifying its use in applications.
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to …
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)H\"older continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent H\"older-stability results for optimal transport maps.
We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called …
We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called Sinkhorn potentials) w.r.t. the regularization parameter, for which we ensure a better than Lipschitz dependence. Such facts may be a first step towards a mathematical justification of annealing or $\varepsilon$-scaling heuristics for the numerical resolution of regularized semi-discrete optimal transport. Our results also entail a non-asymptotic and tight expansion of the difference between the entropic and the unregularized costs.
This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on R^d , which we denote T$\mu$. …
This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on R^d , which we denote T$\mu$. Assuming that the source density $\rho$ is bounded from above and below on a compact convex set, we prove that the map $\mu$ $\rightarrow$ T$\mu$ is bi-H{\"o}lder continuous on large families of probability measures, such as the set of probability measures whose moment of order p > d is bounded by some constant. These stability estimates show that the linearized optimal transport metric W2,$\rho$($\mu$, $\nu$) = T$\mu$ -- T$\nu$ L 2 ($\rho$,R d) is bi-H{\"o}lder equivalent to the 2-Wasserstein distance on such sets, justifiying its use in applications.
We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called …
We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called Sinkhorn potentials) w.r.t. the regularization parameter, for which we ensure a better than Lipschitz dependence. Such facts may be a first step towards a mathematical justification of annealing or $\varepsilon$-scaling heuristics for the numerical resolution of regularized semi-discrete optimal transport. Our results also entail a non-asymptotic and tight expansion of the difference between the entropic and the unregularized costs.
We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this …
We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain.
We study the convergence rate of Sinkhorn's algorithm for solving entropy-regularized optimal transport problems when at least one of the probability measures, $\mu$, admits a density over $\mathbb{R}^d$. For a …
We study the convergence rate of Sinkhorn's algorithm for solving entropy-regularized optimal transport problems when at least one of the probability measures, $\mu$, admits a density over $\mathbb{R}^d$. For a semi-concave cost function bounded by $c_{\infty}$ and a regularization parameter $\lambda > 0$, we obtain exponential convergence guarantees on the dual sub-optimality gap with contraction rate polynomial in $\lambda/c_{\infty}$. This represents an exponential improvement over the known contraction rate $1 - \Theta(\exp(-c_{\infty}/\lambda))$ achievable via Hilbert's projective metric. Specifically, we prove a contraction rate value of $1-\Theta(\lambda^2/c_\infty^2)$ when $\mu$ has a bounded log-density. In some cases, such as when $\mu$ is log-concave and the cost function is $c(x,y)=-\langle x, y \rangle$, this rate improves to $1-\Theta(\lambda/c_\infty)$. The latter rate matches the one that we derive for the transport between isotropic Gaussian measures, indicating tightness in the dependency in $\lambda/c_\infty$. Our results are fully non-asymptotic and explicit in all the parameters of the problem.
We study the convergence rate of Sinkhorn's algorithm for solving entropy-regularized optimal transport problems when at least one of the probability measures, $\mu$, admits a density over $\mathbb{R}^d$. For a …
We study the convergence rate of Sinkhorn's algorithm for solving entropy-regularized optimal transport problems when at least one of the probability measures, $\mu$, admits a density over $\mathbb{R}^d$. For a semi-concave cost function bounded by $c_{\infty}$ and a regularization parameter $\lambda > 0$, we obtain exponential convergence guarantees on the dual sub-optimality gap with contraction rate polynomial in $\lambda/c_{\infty}$. This represents an exponential improvement over the known contraction rate $1 - \Theta(\exp(-c_{\infty}/\lambda))$ achievable via Hilbert's projective metric. Specifically, we prove a contraction rate value of $1-\Theta(\lambda^2/c_\infty^2)$ when $\mu$ has a bounded log-density. In some cases, such as when $\mu$ is log-concave and the cost function is $c(x,y)=-\langle x, y \rangle$, this rate improves to $1-\Theta(\lambda/c_\infty)$. The latter rate matches the one that we derive for the transport between isotropic Gaussian measures, indicating tightness in the dependency in $\lambda/c_\infty$. Our results are fully non-asymptotic and explicit in all the parameters of the problem.
We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this …
We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain.
We study the quantitative stability of the quadratic optimal transport map between a fixed probability density ρ and a probability measure μ on Rd, which we denote Tμ. Assuming that …
We study the quantitative stability of the quadratic optimal transport map between a fixed probability density ρ and a probability measure μ on Rd, which we denote Tμ. Assuming that the source density ρ is bounded from above and below on a compact convex set, we prove that the map μ↦Tμ is bi-Hölder continuous on large families of probability measures, such as the set of probability measures whose moment of order p>d is bounded by some constant. These stability estimates show that the linearized optimal transport metric W2,ρ(μ,ν)=‖Tμ−Tν‖L2(ρ,Rd) is bi-Hölder equivalent to the 2-Wasserstein distance on such sets, justifying its use in applications.
We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called …
We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called Sinkhorn potentials) w.r.t. the regularization parameter, for which we ensure a better than Lipschitz dependence. Such facts may be a first step towards a mathematical justification of annealing or $\varepsilon$-scaling heuristics for the numerical resolution of regularized semi-discrete optimal transport. Our results also entail a non-asymptotic and tight expansion of the difference between the entropic and the unregularized costs.
We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called …
We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called Sinkhorn potentials) w.r.t. the regularization parameter, for which we ensure a better than Lipschitz dependence. Such facts may be a first step towards a mathematical justification of annealing or $\varepsilon$-scaling heuristics for the numerical resolution of regularized semi-discrete optimal transport. Our results also entail a non-asymptotic and tight expansion of the difference between the entropic and the unregularized costs.
This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on R^d , which we denote T$\mu$. …
This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on R^d , which we denote T$\mu$. Assuming that the source density $\rho$ is bounded from above and below on a compact convex set, we prove that the map $\mu$ $\rightarrow$ T$\mu$ is bi-H{\"o}lder continuous on large families of probability measures, such as the set of probability measures whose moment of order p > d is bounded by some constant. These stability estimates show that the linearized optimal transport metric W2,$\rho$($\mu$, $\nu$) = T$\mu$ -- T$\nu$ L 2 ($\rho$,R d) is bi-H{\"o}lder equivalent to the 2-Wasserstein distance on such sets, justifiying its use in applications.
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to …
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)H\older continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent H\older-stability results for optimal transport maps.
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to …
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)H\"older continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent H\"older-stability results for optimal transport maps.