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Brendan Pass

Publications and source records attributed to Brendan Pass.

53 records · Page 3Linked to original sources
arXiv2014

On Khinchine type inequalities for pairwise independent Rademacher random variables

We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ pairwise independent Rademacher random variables. We establish that an analogue of Khintchine's inequality cannot hold in this setting with a constant that is independent of $N$; in fact, we prove that the best constant one can hope for is at least $N^{1/2-1/p}$. Furthermore, we show that this estimate is sharp for exchangeable vectors when $p = 4$. As a fortunate consequence of our work, we obtain similar results for $3$-wise independent vectors.

math.FA↗
arXiv2014

Multi-marginal optimal transport: theory and applications

Over the past five years, multi-marginal optimal transport, a generalization of the well known optimal transport problem of Monge and Kantorovich, has begun to attract considerable attention, due in part to a wide variety of emerging applications. Here, we survey this problem, addressing fundamental theoretical questions including the uniqueness and structure of solutions. The (partial) answers to these questions uncover a surprising divergence from the classical two marginal setting, and reflect a delicate dependence on the cost function. We go one to describe two applications of the multi-marginal problem.

math.AP↗
arXiv2013

Infinite-body optimal transport with Coulomb Cost

We introduce and analyze symmetric infinite-body optimal transport (OT) problems with cost function of pair potential form. We show that for a natural class of such costs, the optimizer is given by the independent product measure all of whose factors are given by the one-body marginal. This is in striking contrast to standard finite-body OT problems, in which the optimizers are typically highly correlated, as well as to infinite-body OT problems with Gangbo-Swiech cost. Moreover, by adapting a construction from the study of exchangeable processes in probability theory, we prove that the corresponding $N$-body OT problem is well approximated by the infinite-body problem. To our class belongs the Coulomb cost which arises in many-electron quantum mechanics. The optimal cost of the Coulombic N-body OT problem as a function of the one-body marginal density is known in the physics and quantum chemistry literature under the name SCE functional, and arises naturally as the semiclassical limit of the celebrated Hohenberg-Kohn functional. Our results imply that in the inhomogeneous high-density limit (i.e. $N\to\infty$ with arbitrary fixed inhomogeneity profile $ρ/N$), the SCE functional converges to the mean field functional. We also present reformulations of the infinite-body and N-body OT problems as two-body OT problems with representability constraints and give a dual characterization of representable two-body measures which parallels an analogous result by Kummer on quantum representability of two-body density matrices.

math-ph↗
arXiv2013

Decoupling of DeGiorgi-type systems via multi-marginal optimal transport

We exhibit a surprising relationship between elliptic gradient systems of PDEs, multi-marginal Monge-Kantorovich optimal transport problem, and multivariable Hardy-Littlewood inequalities. We show that the notion of an orientable elliptic system, conjectured in [6] to imply that (in low dimensions) solutions with certain monotonicity properties are essentially 1-dimensional, is equivalent to the definition of a compatible cost function, known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems [11]. Orientable nonlinearities and compatible cost functions turned out to be also related to submodular functions, which appear in rearrangement inequalities of Hardy-Littlewood type. We use this equivalence to establish a decoupling result for certain solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport.

math.AP↗
arXiv2013

A general condition for Monge solutions in the multi-marginal optimal transport problem

We develop a general condition on the cost function which is sufficient to imply Monge solution and uniqueness results in the multi-marginal optimal transport problem. This result unifies and generalizes several results in the rather fragmented literature on multi-marginal problems. We also provide a systematic way to generate new examples from old ones.

math.AP↗
arXiv2013

N-density representability and the optimal transport limit of the Hohenberg-Kohn functional

We derive and analyze a hierarchy of approximations to the strongly correlated limit of the Hohenberg-Kohn functional. These "density representability approximations" are obtained by first noting that in the strongly correlated limit, N-representability of the pair density reduces to the requirement that the pair density must come from a symmetric N-point density. One then relaxes this requirement to the existence of a representing symmetric k-point density with k < N. The approximate energy can be computed by simulating a fictitious k-electron system. We investigate the approximations by deriving analytically exact results for a 2-site model problem, and by incorporating them into a self-consistent Kohn-Sham calculation for small atoms. We find that the low order representability conditions already capture the main part of the correlations.

physics.chem-ph↗
arXiv2013

Multi-marginal optimal transport on Riemannian manifolds

We study a multi-marginal optimal transportation problem on a Riemannian manifold, with cost function given by the average distance squared from multiple points to their barycenter. Under a standard regularity condition on the first marginal, we prove that the optimal measure is unique and concentrated on the graph of a function over the first variable, thus inducing a Monge solution. This result generalizes McCann's polar factorization theorem on manifolds from 2 to several marginals, in the same sense that a well known result of Gangbo and Swiech generalizes Brenier's polar factorization theorem on $\mathbb{R}^n$.

math.AP↗
arXiv2012

Remarks on the semi-classical Hohenberg-Kohn functional

In this note, we study an optimal transportation problem arising in density functional theory. We derive an upper bound on the semi-classical Hohenberg-Kohn functional derived by Cotar, Friesecke and Klüppelberg (2012) which can be computed in a straightforward way for a given single particle density. This complements a lower bound derived by the aforementioned authors. We also show that for radially symmetric densities the optimal transportation problem arising in the semi-classical Hohenberg-Kohn functional can be reduced to a 1-dimensional problem. This yields a simple new proof of the explicit solution to the optimal transport problem for two particles found by Cotar, Friesecke and Klüppelberg (2012). For more particles, we use our result to demonstrate two new qualitative facts: first, that the solution can concentrate on higher dimensional submanifolds and second that the solution can be non-unique, even with an additional symmetry constraint imposed.

math.OC↗
arXiv2012

Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions

We prove uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of surplus functions; this class arises naturally in multi-agent matching problems in economics. This result generalizes a seminal result of Gangbo and Świȩch on multi-marginal problems. Of particular interest, we show that this also yields a partial generalization of the Gangbo-Świȩch result to manifolds; alternatively, we we can think of this as a partial extension of McCann's theorem for quadratic costs on manifolds to the multi-marginal setting. We also show that the class of surplus functions considered here neither contains, nor is contained in, another class of surpluses studied by the present author, which also generalized Gangbo and Świȩch's result.

math.AP↗
arXiv2012

Optimal transportation with infinitely many marginals

We formulate and study an optimal transportation problem with infinitely many marginals; this is a natural extension of the multi-marginal problem studied by Gangbo and Swiech (1998). We prove results on the existence, uniqueness and characterization of the optimizer, which are natural extensions of the results of Gangbo and Swiech. The proof relies on a relationship between this problem and the problem of finding barycenters in the Wasserstein space, a connection first observed for finitely many marginals by Agueh and Carlier (2011).

math.AP↗
arXiv2012

On a class of optimal transportation problems with infinitely many marginals

We prove existence and uniqueness results for solutions to a class of optimal transportation problems with infinitely many marginals, supported on the real line. We also provide a characterization of the solution with an explicit formula. We then show that this result implies an infinite dimensional rearrangement inequality, and use it to derive upper bounds on solutions to parabolic PDE via the Feynman-Kac formula. This result can be used to provide model independent bounds on the prices of certain derivatives in mathematical finance. As an another application, we obtain refined phase-space bounds in quantum physics.

math.OC↗
arXiv2011

Regularity properties of optimal transportation problems arising in hedonic pricing models

We study a form of optimal transportation surplus functions which arise in hedonic pricing models. We derive a formula for the Ma-Trudinger-Wang curvature of these functions, yielding necessary and sufficient conditions for them to satisfy \textbf{(A3w)}. We use this to give explicit new examples of surplus functions satisfying \textbf{(A3w)}, of the form $b(x,y)=H(x+y)$ where $H$ is a convex function on $\mathbb{R}^n$. We also show that the space of equilibrium contracts in the hedonic pricing model has the maximal possible dimension, a result of potential economic interest.

math.AP↗
arXiv2011

Convexity and multi-dimensional screening for spaces with different dimensions

We study the principal-agent problem. We show that $b$-convexity of the space of products, a condition which appears in a recent paper by Figalli, Kim and McCann \cite{fkm}, is necessary to formulate the problem as a maximization over a convex set. We then show that when the dimension $m$ of the space of types is larger than the dimension $n$ of the space of products, this condition implies that the extra dimensions do not encode independent economic information. When $m$ is smaller than $n$, we show that under $b$-convexity of the space of products, it is always optimal for the principal to offer goods only from a certain prescribed subset. We show that this is equivalent to offering an $m$-dimensional space of goods.

math.OC↗
arXiv2010

Regularity of optimal transportation between spaces with different dimensions

We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target. We demonstrate that if the target is $c$-convex, then the source has a canonical foliation whose co-dimension is equal to the dimension of the target and the problem reduces to an optimal transportation problem between spaces with equal dimensions. If the $c$-convexity condition fails, we do not expect regularity for arbitrary smooth marginals, but, in the case where the source is 2-dimensional and the target is 1 dimensional, we identify sufficient conditions on the marginals and cost to ensure that the optimal map is continuous.

math.AP↗
arXiv2010

Rectifiability of Optimal Transportation Plans

The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

math.AP↗