Results 1  10
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54
Nonembeddability theorems via Fourier analysis
"... Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group ac ..."
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Cited by 42 (9 self)
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Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.
Continuity, curvature, and the general covariance of optimal transportation
"... Abstract. Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ ..."
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Cited by 33 (13 self)
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Abstract. Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via nonnegativity of the sectional curvature of certain nullplanes in a novel but natural pseudoRiemannian geometry which the cost c induces on the product space M × ¯ M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry. Using the pseudoRiemannian structure, we extend Ma, Trudinger and Wang’s conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principal characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and ¯ M of Rn. This maximum principle plays a key role in Loeper’s Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered. 1.
Globalintime weak measure solutions and finitetime aggregation for nonlocal interaction equations
"... Abstract. In this paper, we provide a wellposedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main pheno ..."
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Cited by 19 (8 self)
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Abstract. In this paper, we provide a wellposedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blowup time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite time total collapse of the solution onto a single point, for compactly supported initial measures. Finally, we give conditions on compensation between the attraction at large distances and local repulsion of the potentials to have globalintime confined systems for compactly supported initial data. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations. 1.
Continuity of maps solutions of optimal transportation problems
, 2005
"... In this paper we investigate the continuity of maps solutions of optimal transportation problems. These maps are expressed through the gradient of a potential for which we establish C 1 and C 1,α regularity. Our results hold assuming a condition on the cost function (condition A3 below), that was th ..."
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Cited by 18 (0 self)
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In this paper we investigate the continuity of maps solutions of optimal transportation problems. These maps are expressed through the gradient of a potential for which we establish C 1 and C 1,α regularity. Our results hold assuming a condition on the cost function (condition A3 below), that was the one used for C 2 a priori estimates in [5]. The optimal potential will solve a MongeAmpère equation of the form det(M(x, ∇φ) + D 2 φ) = f where M depends on the cost function. One of the interesting outcome is that under the condition A3, the regularity obtained is better than the one obtained in the case of the ’usual ’ MongeAmpère equation det D 2 φ = f, in particular we will obtain here C 1,α regularity for φ under the condition f ∈ L p,p> n.
Spectral GromovWasserstein Distances for Shape Matching
"... We introduce a spectral notion of distance between shapes and study its theoretical properties. We show that our distance satisfies the properties of a metric on the class of isometric shapes, which means, in particular, that two shapes are at 0 distance if and only if they are isometric. Our constr ..."
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Cited by 15 (1 self)
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We introduce a spectral notion of distance between shapes and study its theoretical properties. We show that our distance satisfies the properties of a metric on the class of isometric shapes, which means, in particular, that two shapes are at 0 distance if and only if they are isometric. Our construction is similar to the recently proposed GromovWasserstein distance, but rather than viewing shapes merely as metric spaces, we define our distance via the comparison of heat kernels. This allows us to relate our distance to previously proposed spectral invariants used for shape comparison, such as the spectrum of the LaplaceBeltrami operator. In addition, the heat kernel provides a natural notion of scale, which is useful for multiscale shape comparison. We also prove a hierarchy of lower bounds for our distance, which provide increasing discriminative power at the cost of increase in computational complexity. 1.
Planar earthmover is not in l1
 In 47th Symposium on Foundations of Computer Science (FOCS
, 2006
"... We show that any L1 embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid {0, 1,..., n} 2 ⊆ R 2 incurs distortion Ω � � log n �. We also use Fourier analytic techniques to construct a simple L1 embedding of this space which has distortion O(lo ..."
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Cited by 12 (2 self)
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We show that any L1 embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid {0, 1,..., n} 2 ⊆ R 2 incurs distortion Ω � � log n �. We also use Fourier analytic techniques to construct a simple L1 embedding of this space which has distortion O(log n). 1
Partial localization, lipid bilayers, and the elastica functional. in prep
, 2006
"... Abstract. Partial localization is the phenomenon of selfaggregation of mass into highdensity structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that thi ..."
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Cited by 12 (7 self)
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Abstract. Partial localization is the phenomenon of selfaggregation of mass into highdensity structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of twodimensional spatial mass densities, exhibits partial localization and displays the ‘solidlike ’ behavior of cell membranes. Specifically, we show that density fields of moderate energy are partially localized, i.e. resemble thin structures. Deviation from a specific uniform thickness, creation of ‘ends’, and the bending of such structures all carry an energy penalty, of different orders in terms of the thickness of the structure. These findings are made precise in a Gammaconvergence result. We prove that a rescaled version of the energy functional converges in the zerothickness limit to a functional that is defined on a class of planar curves. Finiteness of the limit enforces both optimal thickness and nonfracture; if these conditions are met, then the limit value is given by the classical Elastica (bending) energy of the curve.
Effective dynamics using conditional expectations
"... Abstract. The question of coarsegraining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarsegrained variable ξ(x), where x describes the configuration of the system in a highdimensional space R n, and ξ is a smooth ..."
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Cited by 8 (5 self)
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Abstract. The question of coarsegraining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarsegrained variable ξ(x), where x describes the configuration of the system in a highdimensional space R n, and ξ is a smooth function with value in R (typically a reaction coordinate). It is well known that, given a BoltzmannGibbs distribution on x ∈ R n, the equilibrium properties on ξ(x) are completely determined by the free energy. On the other hand, the question of the effective dynamics on ξ(x) is much more difficult to address. Starting from an overdamped Langevin equation on x ∈ R n, we propose an effective dynamics for ξ(x) ∈ R using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarsegrained dynamics. AMS classification scheme numbers: 35B40, 82C31, 60H10Effective dynamics using conditional expectations 2 1.