9 research outputs found
Counterexample to regularity in average-distance problem
No full textThe average-distance problem is to find the best way to approximate (or represent) a given measure μ on RdRd by a one-dimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure μ, minimize
E(Σ)=∫Rdd(x,Σ)dμ(x)+λH1(Σ)
among connected closed sets, Σ , where λ>0λ>0, d(x,Σ)d(x,Σ) is the distance from x to the set Σ , and H1H1 is the one-dimensional Hausdorff measure. Here we provide, for anyd⩾2d⩾2, an example of a measure μ with smooth density, and convex, compact support, such that the global minimizer of the functional is a rectifiable curve which is not C1C1. We also provide a similar example for the constrained form of the average-distance problem.</p
AVERAGE-DISTANCE PROBLEM FOR PARAMETERIZED CURVES
No full textWe consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite compactly supported measure µ, with µ(Rd)> 0 for p ≥ 1 and λ> 0 we consider the functional E(γ) = Rd d(x,Γγ) pdµ(x) + λLength(γ) where γ: I → Rd, I is an interval in R, Γγ = γ(I), and d(x,Γγ) is the distance of x to Γγ. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure H1, and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form of estimates on the total curvature of the minimizers. We prove that for measures µ supported in two dimensions the minimizing curve is injective if p ≥ 2 or if µ has bounded density. This establishes that the minimization over parameterized curves is equivalent to minimizing over embedded curves and thus confirms that the problem has a geometric interpretation
Average-distance problem for parameterized curves
No full textWe consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite compactly supported measure µ, with µ(R d ) > 0 for p ≥ 1 and λ > 0 we consider the functional
E(γ) = Z Rd d(x, Γγ) p dµ(x) + λ Length(γ)
where γ : I → R d , I is an interval in R, Γγ = γ(I), and d(x, Γγ) is the distance of x to Γγ. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure H1 , and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form of estimates on the total curvature of the minimizers. We prove that for measures µ supported in two dimensions the minimizing curve is injective if p ≥ 2 or if µ has bounded density. This establishes that the minimization over parameterized curves is equivalent to minimizing over embedded curves and thus confirms that the problem has a geometric interpretation.</p
Estimating perimeter using graph cuts
No full textAbstract
We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For Ω ⊆ D = (0, 1)d with d ≥ 2, we are given n random independent and identically distributed points on D whose membership in Ω is known. We consider the sample as a random geometric graph with connection distance ε > 0. We estimate the perimeter of Ω (relative to D) by the, appropriately rescaled, graph cut between the vertices in Ω and the vertices in D ∖ Ω. We obtain bias and variance estimates on the error, which are optimal in scaling with respect to n and ε. We consider two scaling regimes: the dense (when the average degree of the vertices goes to ∞) and the sparse one (when the degree goes to 0). In the dense regime, there is a crossover in the nature of the approximation at dimension d = 5: we show that in low dimensions d = 2, 3, 4 one can obtain confidence intervals for the approximation error, while in higher dimensions one can obtain only error estimates for testing the hypothesis that the perimeter is less than a given number.
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Confinement in nonlocal interaction equations
No full textWe investigate some dynamical properties of nonlocal interaction equations. We consider sets of particles interacting pairwise via a potential W, as well as continuum descriptions of such systems. The typical potentials we consider are repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially localized configuration remains localized for all times, regardless of the number of particles or their arrangement. In particular we find sufficient conditions on the potential W for the above "confinement" property to hold. We use the framework of weak measure solutions developed in Carrillo et al. (2011) [2] to provide unified treatment of both particle and continuum systems
GLOBAL-IN-TIME WEAK MEASURE SOLUTIONS AND FINITE-TIME AGGREGATION FOR NONLOCAL INTERACTION EQUATIONS
No full textIn this paper we provide a well-posedness 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 infinite time. We develop an existence theory that enables one to go beyond the blow-up 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. 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
Nonlinear mobility continuity equations and generalized displacement convexity
No full textAbstractWe consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex
NONLOCAL-INTERACTION EQUATIONS ON UNIFORMLY PROX-REGULAR SETS
No full textABSTRACT. We study the well-posedness of a class of nonlocal-interaction equations on general domains Ω ⊂ R d, including nonconvex ones. We show that under mild assumptions on the regularity of domains (uniform prox-regularity), for λ-geodesically convex interaction and external potentials, the nonlocal-interaction equations have unique weak measure solutions. Moreover, we show quantitative estimates on the stability of solutions which quantify the interplay of the geometry of the domain and the convexity of the energy. We use these results to investigate on which domains and for which potentials the solutions aggregate to a single point as time goes to infinity. Our approach is based on the theory of gradient flows in spaces of probability measures. 1
