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In order to compare and interpolate signals, we investigate a Riemannian geometry on the space of signals. The metric allows discontinuous signals and measures both horizontal (thus providing many benefits of the Wasserstein metric) and vertical deformations. Moreover, it allows for signed signals, which overcomes the main deficiency of optimal transportation-based metrics in signal processing. We characterize the metric properties of the space of signals and establish the regularity and stability of geodesics. Furthermore, we introduce an efficient numerical scheme to compute the geodesics and present several experiments which highlight the nature of the metric. [ABSTRACT FROM AUTHOR]

Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liu et al 2023 Appl. Math. Optim. 87 48; Lu et al 2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or by χ 2 divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth–death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker–Planck equation and relies on kernel-based approximations of the measure. Using the technique of Γ-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth–death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure. [ABSTRACT FROM AUTHOR]

We study the approximation of the nonlocal-interaction equation restricted to a compact manifold embedded in , and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on can be approximated by the classical nonlocal-interaction equation on by adding an external potential which strongly attracts to. The proof relies on the Sandier–Serfaty approach [ 23 , 24 ] to the -convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on , which was shown [ 10 ]. We also provide an another approximation to the interaction equation on , based on iterating approximately solving an interaction equation on and projecting to. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics. [ABSTRACT FROM AUTHOR]

We consider adaptations of the Mumford–Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford–Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford–Shah functionals converge to a minimizer of a continuum Mumford–Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate minimizers of the graph Mumford–Shah functional. [ABSTRACT FROM AUTHOR]

We investigate a family of regression problems in a semisupervised setting. The task is to assign real-valued labels to a set of n sample points provided a small training subset of N labeled points. A goal of semisupervised learning is to take advantage of the (geometric) structure provided by the large number of unlabeled data when assigning labels. We consider random geometric graphs, with connection radius ε(n), to represent the geometry of the data set. Functionals which model the task reward the regularity of the estimator function and impose or reward the agreement with the training data. Here we consider discrete p-Laplacian regularization. We investigate asymptotic behavior when the number of unlabeled points increases while the number of training points remains fixed. A delicate interplay between the regularizing nature of the functionals and the nonlocality inherent to the graph constructions is uncovered. Rigorous, almost optimal, ranges on the scaling of ε(n) for asymptotic consistency are obtained and in these admissible ranges it is shown that mini-mizers of the discrete functionals converge uniformly to the desired continuum limit. Furthermore, we discover that, for the standard model, there is a restrictive upper bound on how quickly ε(n) must converge to zero as n → ∞. A new model is introduced, which is as simple as the original model, but overcomes this restriction. [ABSTRACT FROM AUTHOR]

Angle-preserving or conformal surface parameterization has proven to be a powerful tool across applications ranging from geometry processing, to digital manufacturing, to machine learning, yet conformal maps can still suffer from severe area distortion. Cone singularities provide a way to mitigate this distortion, but finding the best configuration of cones is notoriously difficult. This paper develops a strategy that is globally optimal in the sense that it minimizes total area distortion among all possible cone configurations (number, placement, and size) that have no more than a fixed total cone angle. A key insight is that, for the purpose of optimization, one should not work directly with curvature measures (which naturally represent cone configurations), but can instead apply Fenchel-Rockafellar duality to obtain a formulation involving only ordinary functions. The result is a convex optimization problem, which can be solved via a sequence of sparse linear systems easily built from the usual cotangent Laplacian. The method supports user-defined notions of importance, constraints on cone angles (e.g., positive, or within a given range), and sophisticated boundary conditions (e.g., convex, or polygonal). We compare our approach to previous techniques on a variety of challenging models, often achieving dramatically lower distortion, and demonstrating that global optimality leads to extreme robustness in the presence of noise or poor discretization. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Transport-based distances, such as the Wasserstein distance and earth mover's distance, have been shown to be an effective tool in signal and image analysis. The success of transport-based distances is in part due to their Lagrangian nature which allows it to capture the important variations in many signal classes. However, these distances require the signal to be non-negative and normalised. Furthermore, the signals are considered as measures and compared by redistributing (transporting) them, which does not directly take into account the signal intensity. Here, we study a transport-based distance, called the $$TL^p$$ distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multichannelled signals. The distance can be computed by existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multichannelled non-positive one-dimensional signals and two-dimensional images, and colour transfer. [ABSTRACT FROM AUTHOR]

We 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) + λ Length(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. [ABSTRACT FROM AUTHOR]

We study well-posedness of a class of nonlocal interaction equations with spatially dependent mobility. We also allow for the presence of boundaries and external potentials. Such systems lead to the study of nonlocal interaction equations on subsets ℳ of ℝdendowed with a Riemannian metricg. We obtain conditions, relating the interaction potential and the geometry, which imply existence, uniqueness and stability of solutions. We study the equations in the setting of gradient flows in the space of probability measures on ℳ endowed with Riemannian 2-Wasserstein metric. [ABSTRACT FROM PUBLISHER]

Given a finite measure μ ≥ 0 with compact support, and γ > 0, the average-distance problem, in the penalized formulation, is to minimize Eμλ (S) := ∫Rd d(x, Σ)dμ(x) + γH1(Σ) among pathwise connected, closed sets, Σ. Here d(x,Σ) is the distance from a point to a set and H¹ is the 1-Hausdorff measure. In a sense the problem is to find a one-dimensional measure that best approximates μ. The minimizer, Σ, is topologically a tree whose branches are rectifiable curves. The branches may not be C¹, even for measures μ with smooth density. Here we show a result on weak second-order regularity of branches. Namely, we show that arc-length-parameterized branches have BV derivatives and provide a priori estimates on the BV norm. The technique we use is to approximate the measure μ, in the weak-* topology of measures, by discrete measures. Such approximation is also relevant for numerical computations. We prove the stability of the minimizers in appropriate spaces and also compare the topologies of the minimizers corresponding to the approximations with the minimizer corresponding to μ and obtain a topological lower-semicontinuity result. [ABSTRACT FROM AUTHOR]

We consider coarsening in interfacial systems driven by nonlocal energies. Of particular interest are the nonlocal Cahn--Hilliard equation and models of biological aggregation. The energies considered cause the system to separate into phases. The pattern of interfaces evolves under nonlocal surface-tension-type effects. The typical length scales grow and the pattern coarsens. We prove a rigorous upper bound on the coarsening rate. The proof uses the energy-based approach to estimates on rate of coarsening introduced by Kohn and Otto [Comm. Math. Phys., 229 (2002), pp. 375--395]. To show the required estimates on the flatness of the energy landscape we develop a geometric approach which is applicable to a wider class of problems, which includes ones based on local, gradient-type energies. [ABSTRACT FROM AUTHOR]

Certain liquids on solid substrates form a configuration of droplets connected by a precursor layer. This configuration coarsens: The average droplet size grows while the number of droplets decreases and the characteristic distance between them increases. We study this type of coarsening behavior in a model given by an evolution equation for the film height on an n-dimensional substrate. Heuristic arguments based on the asymptotic analysis of Glasner and Witelski [Phys. Rev. E, 67 (2003), p. 016302, Phys. D., 209 (2005), pp. 80-104] and numerical simulations suggest a statistically self-similar behavior characterized by a single exponent which determines the coarsening rate. In this paper, we establish rigorously an upper bound on the coarsening rate in a time-averaged sense. We use the fact that the evolution is a gradient flow, i.e., a steepest descent in an energy landscape. Coarse information on the geometry of the energy landscape serves to obtain coarse information on the dynamics. This robust method was proposed in [R. V. Kohn and F. Otto, Comm. Math. Phys., 229 (2002), pp. 375-395]. Our main analytical contribution is an interpolation inequality involving the Wasserstein distance, which characterizes the coarse shape of the energy landscape. The upper bound we obtain is in agreement with heuristic arguments and numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior as ℇ→0 of the solutions of nonlocal reaction–diffusion equations of the form ut-Δu+ℇ-2f(u,ℇ∫0u)=0 in O×(0,T) associated with nonlinear oblique derivative boundary conditions. We show that such behavior is described in terms of an interface evolving with normal velocity depending not only on its curvature but also on the measure of the set it encloses. To this purpose we introduce a weak notion of motion of hypersurfaces with nonlocal normal velocities depending on the volume they enclose, which extends the geometric definition of generalized motion of hypersurfaces in bounded domains introduced by G. Barles and the first author to solve a similar problem with local normal velocities depending on the normal direction and the curvature of the front. We also establish comparison and existence theorems of viscosity solutions to initial‐boundary value problems for some singular degenerate nonlocal parabolic pde's with nonlinear Neumann‐type boundary conditions. [ABSTRACT FROM AUTHOR]

The convergence of schemes for propagation of fronts in a bounded domain moving with normal velocities is studied. The velocities considered depend on the principal curvatures, the normal direction, the location, as well as some nonlocal properties of the front. Most of the schemes considered are in essence threshold dynamics type approximation schemes, modified for Neumann boundary conditions and nonlocal terms. The existence and uniqueness of appropriately defined viscosity solutions of the level-set equations describing the nonlocal motions is also shown. [Copyright &y& Elsevier]