arXiv subjects

Brendan Pass

Publications and source records attributed to Brendan Pass.

At least 37 records · Page 2Linked to original sources

arXiv2022

Monge solutions and uniqueness in multi-marginal optimal transport: weaker conditions on the cost, stronger conditions on the marginals

We establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to local coordinates. When only the first marginal is assumed to be absolutely continuous, our condition is equivalent to the twist on splitting sets condition found in [23]. In addition, it is satisfied by the special cost functions in our earlier work [32, 33], when absolute continuity is imposed on certain other collections of marginals. We also present several new examples of cost functions which violate the twist on splitting sets condition but satisfy the new condition introduced here; we therefore obtain Monge solution and uniqueness results for these cost functions, under regularity conditions on an appropriate subset of the marginals.

math.OC↗

arXiv2021

Monge solutions and uniqueness in multi-marginal optimal transport via graph theory

We study a multi-marginal optimal transport problem with surplus $b(x_{1}, \ldots, x_{m})=\sum_{\{i,j\}\in P} x_{i}\cdot x_{j}$, where $P\subseteq Q:=\{\{i,j\}: i, j \in \{1,2,...m\}, i \neq j\}$. We reformulate this problem by associating each surplus of this type with a graph with $m$ vertices whose set of edges is indexed by $P$. We then establish uniqueness and Monge solution results for two general classes of surplus functions. Among many other examples, these classes encapsulate the Gangbo and Świȩch surplus [12] and the surplus $\sum_{i=1}^{m-1}x_{i}\cdot x_{i+1} + x_{m}\cdot x_{1}$ studied in an earlier work by the present authors [23].

math.OC↗

arXiv2021

Identification of hedonic equilibrium and nonseparable simultaneous equations

This paper derives conditions under which preferences and technology are nonparametrically identified in hedonic equilibrium models, where products are differentiated along more than one dimension and agents are characterized by several dimensions of unobserved heterogeneity. With products differentiated along a quality index and agents characterized by scalar unobserved heterogeneity, single crossing conditions on preferences and technology provide identifying restrictions in Ekeland, Heckman and Nesheim (2004) and Heckman, Matzkin and Nesheim (2010). We develop similar shape restrictions in the multi-attribute case. These shape restrictions, which are based on optimal transport theory and generalized convexity, allow us to identify preferences for goods differentiated along multiple dimensions, from the observation of a single market. We thereby derive nonparametric identification results for nonseparable simultaneous equations and multi-attribute hedonic equilibrium models with (possibly) multiple dimensions of unobserved heterogeneity. One of our results is a proof of absolute continuity of the distribution of endogenously traded qualities, which is of independent interest.

econ.EM↗

arXiv2020

Multi-marginal optimal transport on the Heisenberg group

We consider the multi-marginal optimal transport of aligning several compactly supported marginals on the Heisenberg group to minimize the total cost, which we take to be the sum of the squared Carnot-Carathéodory distances from the marginal points to their barycenter. Under certain technical hypotheses, we prove existence and uniqueness of optimal maps. We also point out several related open questions.

math.OC↗

arXiv2020

Generalized barycenters and variance maximization on metric spaces

We show that the variance of a probability measure $μ$ on a compact subset $X$ of a complete metric space $M$ is bounded by the square of the circumradius $R$ of the canonical embedding of $X$ into the space $P(M)$ of probability measures on $M$, equipped with the Wasserstein metric. When barycenters of measures on $X$ are unique (such as on CAT($0$) spaces), our approach shows that $R$ in fact coincides with the circumradius of $X$ and so this result extends a recent result of Lim-McCann from Euclidean space. Our approach involves bi-linear minimax theory on $P(X) \times P(M)$ and extends easily to the case when the variance is replaced by very general moments. As an application, we provide a simple proof of Jung's theorem on CAT($k$) spaces, a result originally due to Dekster and Lang-Schroeder.

math.PR↗

arXiv2020

Multi-marginal optimal transportation problem for cyclic costs

We study a multi-marginal optimal transportation problem with a cost function of the form $c(x_{1}, \ldots,x_{m})=\sum_{k=1}^{m-1}|x_{k}-x_{k+1}|^{2} + |x_{m}- F(x_{1})|^{2}$, where $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$. When $m=4$, $F$ is a positive multiple of the identity mapping, and the first and last marginals are absolutely continuous with respect to Lebesgue measure, we establish that any solution of the Kantorovich problem is induced by a map; the solution is therefore unique. We go on to show that this result is sharp in a certain sense. Precisely, we exhibit examples showing that Kantorovich solutions may concentrate on a higher dimensional sets if any of the following hold: 1) $F$ is any linear mapping other than a positive scalar multiple of the identity, 2) the last marginal is not absolutely continuous with respect to Lebesgue measure, or 3) the number of marginals $m \geq 5$, even when $F$ is the identity mapping.

math.OC↗

arXiv2019

Variational problems involving unequal dimensional optimal transport

This paper is devoted to variational problems on the set of probability measures which involve optimal transport between unequal dimensional spaces. In particular, we study the minimization of a functional consisting of the sum of a term reflecting the cost of (unequal dimensional) optimal transport between one fixed and one free marginal, and another functional of the free marginal (of various forms). Motivating applications include Cournot-Nash equilibria where the strategy space is lower dimensional than the space of agent types. For a variety of different forms of the term described above, we show that a nestedness condition, which is known to yield much improved tractability of the optimal transport problem, holds for any minimizer. Depending on the exact form of the functional, we exploit this to find local differential equations characterizing solutions, prove convergence of an iterative scheme to compute the solution, and prove regularity results.

math.AP↗

arXiv2019

Optimal transport and barycenters for dendritic measures

We introduce and study a variant of the Wasserstein distance on the space of probability measures, specially designed to deal with measures whose support has a dendritic, or treelike structure with a particular direction of orientation. Our motivation is the comparison of and interpolation between plants' root systems. We characterize barycenters with respect to this metric, and establish that the interpolations of root-like measures, using this new metric, are also root like, in a certain sense; this property fails for conventional Wasserstein barycenters. We also establish geodesic convexity with respect to this metric for a variety of functionals, some of which we expect to have biological importance.

math.OC↗

arXiv2019

Optimal transportation between unequal dimensions

We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Ampère equation, introduced here for the first time. Under suitable topological conditions, we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution. We show that this local equation is elliptic, and $C^{2,α}$ solutions can therefore be bootstrapped to obtain higher regularity results, assuming smoothness of the corresponding differential operator, which we prove under simplifying assumptions. For one-dimensional targets, our sufficient criteria for regularity of solutions to the resulting ODE are considerably less restrictive than those required by earlier works.

math.AP↗

arXiv2017

On Khintchine type inequalities for $k$-wise independent Rademacher random variables

We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ $k$-wise independent Rademacher random variables. We show that an analogue of Khintchine's inequality holds, with a constant $N^{1/2-k/2p}$, when $k$ is even. We then show that this result is sharp for $k=2$; in particular, a version of Khintchine's inequality for sequences of pairwise Rademacher variables \emph{cannot} hold with a constant independent of $N$. We also characterize the cases of equality and show that, although the vector achieving equality is not unique, it is unique (up to law) among the smaller class of exchangable vectors of pairwise independent Rademacher random variables. As a fortunate consequence of our work, we obtain similar results for $3$-wise independent vectors.

math.PR↗

arXiv2017

A canonical barycenter via Wasserstein regularization

We introduce a weak notion of barycenter of a probability measure $μ$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter $B(μ)$ is well defined; it is a probability measure on $X$ supported on the set of the usual metric barycenter points of the given measure $μ$. The definition uses the canonical embedding of the metric space $X$ into its Wasserstein space $P(X)$, pushing a given measure $μ$ forward to a measure on $P(X)$. We then regularize the measure by the Wasserstein distance to the reference measure ${\bf m}$, and obtain a uniquely defined measure on $X$ supported on the barycentric points of $μ$. We investigate various properties of $B(μ)$

math.OC↗

arXiv2017

Interpolating between matching and hedonic pricing models

We consider the theoretical properties of a model which encompasses bi-partite matching under transferable utility on the one hand, and hedonic pricing on the other. This framework is intimately connected to tripartite matching problems (known as multi-marginal optimal transport problems in the mathematical literature). We exploit this relationship in two ways; first, we show that a known structural result from multi-marginal optimal transport can be used to establish an upper bound on the dimension of the support of stable matchings. Next, assuming the distribution of agents on one side of the market is continuous, we identify a condition on their preferences that ensures purity and uniqueness of the stable matching; this condition is a variant of a known condition in the mathematical literature, which guarantees analogous properties in the multi-marginal optimal transport problem. We exhibit several examples of surplus functions for which our condition is satisfied, as well as some for which it fails.

econ.GN↗

arXiv2016

Multidimensional matching

We present a general analysis of multidimensional matching problems with transferable utility, paying particular attention to the case in which the dimensions of heterogeneity on the two sides of the market are unequal. A particular emphasis is put on problems where agents on one side of the market are multidimensional and agents on the other side are uni-dimensional, we describe a general approach to solve such problems. Lastly, we analyze several examples, including an hedonic model with differentiated products, a marriage market model where wives are differentiated in income and fertility, and a competitive variation of the Rochet-Choné problem. In the latter example, we show that the bunching phenomena, observed by Rochet and Choné in the monopoly context, do not occur in the competitive context

econ.GN↗

arXiv2016

Multi- to one-dimensional transportation

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 $s \in C^{1,1}$ which ensures the set of $x$ paired with [$g$-a.e.] $y\in Y$ lie in a codimension $n$ submanifold of $X$. Specializing to the case $m>n=1$, we discover a nestedness criteria relating $s$ to $(f,g)$ which allows us to construct a unique optimal solution in the form of a map $F:X \longrightarrow \overline Y$. When $s\in C^2 \cap W^{3,1}$ and $\log f$ and $\log g$ are bounded, the Kantorovich dual potentials $(u,v)$ satisfy $v \in C^{1,1}_{loc}(Y)$, and the normal velocity $V$ of $F^{-1}(y)$ with respect to changes in $y$ is given by $V(x) = v"(f(x))-s_{yy}(x,f(x))$. Positivity (b) of $V$ locally implies a Lipschitz bound on $f$; moreover, $v \in C^2$ if ${F^{-1}(y)}$ intersects $\partial X \in C^1$ transversally (c). On subsets where (a)-(c) can be be quantified, for each integer $r \ge1$ the norms of $u,v \in C^{r+1,1}$ and $F \in C^{r,1}$ are controlled by these bounds, $||\log f,\log g, \partial X ||_{C^{r-1,1}}, ||\partial X||_{C^{1,1}}$, $||s||_{C^{r+1,1}}$, and the smallness of $F^{-1}(y)$. We give examples showing regularity extends from $X$ to part of $\bar X$, but not from $Y$ to $\bar Y$. We also show that when $s$ remains nested for all $(f,g)$, the problem in $\mathbf{R}^m \times \mathbf{R}$ reduces to a supermodular problem in $\mathbf{R} \times \mathbf{R}$.

math.AP↗

arXiv2015

Wasserstein Barycenters over Riemannian manifolds

We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures $Ω\in P(P(M))$ on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier \cite{ac} to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschläger \cite{c-ems} for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where $Ω$ is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn-Minkowski inequality for a random measurable set on a Riemannian manifold.

math.AP↗

arXiv2015

The multi-marginal optimal partial transport problem

We introduce and study a multi-marginal optimal partial transport problem. Under a natural and sharp condition on the dominating marginals, we establish uniqueness of the optimal plan. Our strategy of proof establishes and exploits a connection with another novel problem, which we call the Monge-Kantorovich partial barycenter problem (with quadratic cost). This latter problem has a natural interpretation as a variant of the mines and factories description of optimal transport. We then turn our attention to various analytic properties of these two problems. Of particular interest, we show that monotonicity of the active marginals can fail, a surprising difference from the two marginal case.

math.AP↗

arXiv2015

Solutions to multi-marginal optimal transport problems concentrated on several graphs

We study solutions to the multi-marginal Monge-Kantorovich problem which are concentrated on several graphs over the first marginal. We first present two general conditions on the cost function which ensure, respectively, that any solution must concentrate on either finitely many or countably many graphs. We show that local differential conditions on the cost, known to imply local $d$-rectifiability of the solution, are sufficient to imply a local version of the first of our conditions. We exhibit two examples of cost functions satisfying our conditions, including the Coulomb cost from density functional theory in one dimension. We also prove a number of results relating to the uniqueness and extremality of optimal measures. These include a sufficient condition on a collection of graphs for any competitor in the Monge-Kantorovich problem concentrated on them to be extremal, and a general negative result, which shows that when the problem is symmetric with respect to permutations of the variables, uniqueness cannot occur except under very special circumstances.

math.OC↗

arXiv2015

Nonpositive curvature, the variance functional, and the Wasserstein barycenter

This paper connects nonpositive sectional curvature of a Riemannian manifold with the displacement convexity of the variance functional on the space $P(M)$ of probability measures over $M$. We show that $M$ has nonpositive sectional curvature and has trivial topology (i.e, is homeomorphic to $\mathbf{R}^n$) if and only if the variance functional on $P(M)$ is displacement convex. This is followed by a Jensen type inequality for the variance functional with respect to Wasserstein barycenters, as well as by a result comparing the variance of the Wasserstein and linear barycenters of a probability measure on $P(M)$ (that is, an element of $P(P(M))$). These results are applied to invariant measures under isometry group actions, giving a comparison for the variance functional between the Wasserstein projection and the $L^2$ projection to the set of invariant measures.

math.DG↗