We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings mu : Omega -> (P(D), W ) defined over a subset Omega of R^p and valued in the space P(D) of probability measures on a compact convex subset D of R^q endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetnyak of Sobolev spaces valued in metric spaces. We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary d Omega are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for Omega a segment of R. As the Wasserstein space (P(D), W ) is positively curved in the sense of Alexandrov we cannot apply the theory of Koorevaar, Schoen and Jost and we use instead arguments based on optimal transport. We manage to get existence of harmonic mappings provided that the boundary values are Lipschitz on d Omega, uniqueness is an open question. If Omega is a segment of R, it is known that a curve valued in the Wasserstein space P(D) can be seen as a superposition of curves valued in D. We show that it is no longer the case in higher dimensions: a generic mapping Omega -> P(D) cannot be represented as the superposition of mappings Omega -> D. We are able to show a Ishihara-type property: the composition F(mu) of a function F : P(D) -> R convex along generalized geodesics and a harmonic mapping mu : Omega -> P(D) is a subharmonic real-valued function. We also study the special case where we restrict ourselves to a given family of elliptically contoured distributions (a finite-dimensional and geodesically convex submanifold of (P(D), W ) which generalizes the case of Gaussian measures) and show that it boils down to harmonic mappings valued in the Riemannian manifold of symmetric matrices endowed with the distance coming from optimal transport.
Finally, in @cite_2 the authors also study mappings valued in the space of probability measures, but are rather interested in the bounded variation norm (the integral of the norm of the gradient) than in the Dirichlet energy. Their provide applications to the denoising of measure-valued images.
@cite_2
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These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. Soft maps taking points on one surface to probability distributions on another are attractive for representing surface mappings in the presence of symmetry, ambiguity, and combinatorial complexity. Few techniques, however, are available to measure their continuity and other properties. To this end, we introduce a novel Dirichlet energy for soft maps generalizing the classical map Dirichlet energy, which measures distortion by computing how soft maps transport probabilistic mass from one distribution to another. We formulate the computation of the Dirichlet energy in terms of a differential equation and provide a finite elements discretization that enables all of the quantities introduced to be computed. We demonstrate the effectiveness of our framework for understanding soft maps arising from various sources. Furthermore, we suggest how these energies can be applied to generate continuous soft or point-to-point maps. We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in "subdomains") that directly lends itself to operator preconditioning via Calderon projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise @math -distributions over each associated interface. Through the use of interior Calderon projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions' projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning. In this paper we address the problem of dense depth map estimation from sparse noisy range data to reconstruct large heterogeneous outdoor scenes. We propose a surface inpainting solution through energy minimisation with an adaptive selection of surface regularisers among a set of well known convex and non-convex regularisers. In fact, the selection of norm is pivotal with respect to the intrinsic surface characteristics. Our goal is to show how dense interpolation of sparse range data can be leveraged of more exotic and non-convex regularisers such as the log and logTGV [1] which can better capture the scene geometry. In contrast to state of the art solutions, we do not restrict ourselves to this set of norms, instead we search for the most apt norm for each semantically segmented part of the scene. Our energy model selection use Bayesian optimisation to learn the best choice of free parameters. This results in an adaptive model selection and the generalisation of well studied regularisation norms. We conclude with a detailed experimental analysis of our approach using a basis of four norms over a set of challenging outdoor scenes. We develop a general mathematical framework for variational problems where the unknown function assumes values in the space of probability measures on some metric space. We study weak and strong topologies and define a total variation seminorm for functions taking values in a Banach space. The seminorm penalizes jumps and is rotationally invariant under certain conditions. We prove existence of a minimizer for a class of variational problems based on this formulation of total variation, and provide an example where uniqueness fails to hold. Employing the Kan -torovich-Rubinstein transport norm from the theory of optimal transport, we propose a variational approach for the restoration of orientation distribution function (ODF)-valued images, as commonly used in Diffusion MRI. We demonstrate that the approach is numerically feasible on several data sets. Abstract Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles yield stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4 M − 4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M = 2 , 3 . Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer–Rao lower bounds to identify stability with stronger versions of the injectivity characterizations. Abstract We investigate the recovery of almost s -sparse vectors x ∈ C N from undersampled and inaccurate data y = A x + e ∈ C m by means of minimizing ‖ z ‖ 1 subject to the equality constraints A z = y . If m ≍ s ln ( N s ) and if Gaussian random matrices A ∈ R m × N are used, this equality-constrained l 1 -minimization is known to be stable with respect to sparsity defects and robust with respect to measurement errors. If m ≍ s ln ( N s ) and if Weibull random matrices are used, we prove here that the equality-constrained l 1 -minimization remains stable and robust. The arguments are based on two key ingredients, namely the robust null space property and the quotient property. The robust null space property relies on a variant of the classical restricted isometry property where the inner norm is replaced by the l 1 -norm and the outer norm is replaced by a norm comparable to the l 2 -norm. For the l 1 -minimization subject to inequality constraints, this yields stability and robustness results that are also valid when considering sparsity relative to a redundant dictionary. As for the quotient property, it relies on lower estimates for the tail probability of sums of independent Weibull random variables. We investigate recovery of nonnegative vectors from non-adaptive compressive measurements in the presence of noise of unknown power. In the absence of noise, existing results in the literature identify properties of the measurement that assure uniqueness in the non-negative orthant. By linking such uniqueness results to nullspace properties, we deduce uniform and robust compressed sensing guarantees for nonnegative least squares. No l1-regularization is required. As an important proof of principle, we establish that m x n random i.i.d. 0 1-valued Bernoulli matrices obey the required conditions with overwhelming probability provided that m=O(slog(n s)). We achieve this by establishing the robust nullspace property for random 0 1-matrices - a novel result in its own right. Our analysis is motivated by applications in wireless network activity detection. Given topological spaces X1, ..., Xn with product space X, probability measures μi on Xi together with a real function h on X define a marginal problem as well as a dual problem. Using an extended version of Choquet's theorem on capacities, an analogue of the classical duality theorem of linear programming is established, imposing only weak conditions on the topology of the spaces Xi and the measurability resp. boundedness of the function h. Applications concern, among others, measures with given support, stochastic order and general marginal problems. 0. Background Notes.- 1. Continuous Partitions of Unity.- 2. Absolutely Continuous Functions.- 3. Some Compactness Theorems.- 4. Weak Convergence and Asymptotic Center of Bounded Sequences.- 5. Closed Convex Hulls and the Mean-Value Theorem.- 6. Lower Semicontinuous Convex Functions and Projections of Best Approximation.- 7. A Concise Introduction to Convex Analysis.- 1. Set-Valued Maps.- 1. Set-Valued Maps and Continuity Concepts.- 2. Examples of Set-Valued Maps.- 3. Continuity Properties of Maps with Closed Convex Graph.- 4. Upper Hemicontinuous Maps and the Convergence Theorem.- 5. Hausdorff Topology.- 6. The Selection Problem.- 7. The Minimal Selection.- 8. Chebishev Selection.- 9. The Barycentric Selection.- 10. Selection Theorems for Locally Selectionable Maps.- 11. Michael's Selection Theorem.- 12. The Approximate Selection Theorem and Kakutani's Fixed Point Theorem.- 13. (7-Selectionable Maps.- 14. Measurable Selections.- 2. Existence of Solutions to Differential Inclusions.- 1. Convex Valued Differential Inclusions.- 2. Qualitative Properties of the Set of Trajectories of Convex-Valued Differential Inclusions.- 3. Nonconvex-Valued Differential Inclusions.- 4. Differential Inclusions with Lipschitzean Maps and the Relaxation Theorem.- 5. The Fixed-Point Approach.- 6. The Lower Semicontinuous Case.- 3. Differential Inclusions with Maximal Monotone Maps.- 1. Maximal Monotone Maps.- 2. Existence and Uniqueness of Solutions to Differential Inclusions with Maximal Monotone Maps.- 3. Asymptotic Behavior of Trajectories and the Ergodic Theorem.- 4. Gradient Inclusions.- 5. Application: Gradient Methods for Constrained Minimization Problems.- 4. Viability Theory: The Nonconvex Case.- 1. Bouligand's Contingent Cone.- 2. Viable and Monotone Trajectories.- 3. Contingent Derivative of a Set-Valued Map.- 4. The Time Dependent Case.- 5. A Continuous Version of Newton's Method.- 6. A Viability Theorem for Continuous Maps with Nonconvex Images..- 7. Differential Inclusions with Memory.- 5. Viability Theory and Regulation of Controled Systems: The Convex Case.- 1. Tangent Cones and Normal Cones to Convex Sets.- 2. Viability Implies the Existence of an Equilibrium.- 3. Viability Implies the Existence of Periodic Trajectories.- 4. Regulation of Controled Systems Through Viability.- 5. Walras Equilibria and Dynamical Price Decentralization.- 6. Differential Variational Inequalities.- 7. Rate Equations and Inclusions.- 6. Liapunov Functions.- 1. Upper Contingent Derivative of a Real-Valued Function.- 2. Liapunov Functions and Existence of Equilibria.- 3. Monotone Trajectories of a Differential Inclusion.- 4. Construction of Liapunov Functions.- 5. Stability and Asymptotic Behavior of Trajectories.- Comments. Invariance properties of random walks in bounded domains are a topic of growing interest since they contribute to improving our understanding of diffusion in confined geometries. Recently, limited to Pearson random walks with exponentially distributed straight paths, it has been shown that under isotropic uniform incidence, the average length of the trajectories through the domain is independent of the random walk characteristic and depends only on the ratio of the volume's domain over its surface. In this paper, thanks to arguments of integral geometry, we generalize this property to any isotropic bounded stochastic process and we give the conditions of its validity for isotropic unbounded stochastic processes. The analytical form for the traveled distance from the boundary to the first scattering event that ensures the validity of the Cauchy formula is also derived. The generalization of the Cauchy formula is an analytical constraint that thus concerns a very wide range of stochastic processes, from the original Pearson random walk to a Rayleigh distribution of the displacements, covering many situations of physical importance. We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from ℝ 2+1 Minkowski space into the sphere S 2 . We establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities. The construction and analysis are done in the context of the k-equivariant symmetry reduction, and we restrict to maps with homotopy class k ≥ 4. The concentration mechanism we uncover is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. We show that the phenomenon is generic (e.g. in certain Sobolev spaces) in that it persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic. In this paper we investigate the use of Langevin Monte Carlo methods on the probability simplex and propose a new method, Stochastic gradient Riemannian Langevin dynamics, which is simple to implement and can be applied to large scale data. We apply this method to latent Dirichlet allocation in an online mini-batch setting, and demonstrate that it achieves substantial performance improvements over the state of the art online variational Bayesian methods. A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings: we denote this as γk, indexed by the kernel function k that defines the inner product in the RKHS. We present three theoretical properties of γk. First, we consider the question of determining the conditions on the kernel k for which γk is a metric: such k are denoted characteristic kernels. Unlike pseudometrics, a metric is zero only when two distributions coincide, thus ensuring the RKHS embedding maps all distributions uniquely (i.e., the embedding is injective). While previously published conditions may apply only in restricted circumstances (e.g., on compact domains), and are difficult to check, our conditions are straightforward and intuitive: integrally strictly positive definite kernels are characteristic. Alternatively, if a bounded continuous kernel is translation-invariant on ℜd, then it is characteristic if and only if the support of its Fourier transform is the entire ℜd. Second, we show that the distance between distributions under γk results from an interplay between the properties of the kernel and the distributions, by demonstrating that distributions are close in the embedding space when their differences occur at higher frequencies. Third, to understand the nature of the topology induced by γk, we relate γk to other popular metrics on probability measures, and present conditions on the kernel k under which γk metrizes the weak topology. We show that heterogeneous degree distributions in observed scale-free topologies of complex networks can emerge as a consequence of the exponential expansion of hidden hyperbolic space. Fermi-Dirac statistics provides a physical interpretation of hyperbolic distances as energies of links. The hidden space curvature affects the heterogeneity of the degree distribution, while clustering is a function of temperature. We embed the internet into the hyperbolic plane and find a remarkable congruency between the embedding and our hyperbolic model. Besides proving our model realistic, this embedding may be used for routing with only local information, which holds significant promise for improving the performance of internet routing. We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the otherby solving Poisson's orHelmholtz's equation with well-chosen boundar y conditions. Using this equivalence, we study the case of a large class of region functionals by standard methods of the calculus of variations and derive the corresponding Euler-Lagrange equations. We revisit this problem using the notion of a shape derivative and show that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals. We also define a larger class of region functionals based on the estimation and comparison to a prototype of the probability density distribution of image features and show how the shape derivative tool allows us to easily compute the corresponding Gateaux derivatives and Euler-Lagrange equations. Finally we apply this new functional to the problem of regions segmentation in sequences of color images. We briefly describe our numerical scheme and show some experimental results. In uncertain scalar fields where data values vary with a certain probability, the strength of this variability indicates the confidence in the data. It does not, however, allow inferring on the effect of uncertainty on differential quantities such as the gradient, which depend on the variability of the rate of change of the data. Analyzing the variability of gradients is nonetheless more complicated, since, unlike scalars, gradients vary in both strength and direction. This requires initially the mathematical derivation of their respective value ranges, and then the development of effective analysis techniques for these ranges. This paper takes a first step into this direction: Based on the stochastic modeling of uncertainty via multivariate random variables, we start by deriving uncertainty parameters, such as the mean and the covariance matrix, for gradients in uncertain discrete scalar fields. We do not make any assumption about the distribution of the random variables. Then, for the first time to our best knowledge, we develop a mathematical framework for computing confidence intervals for both the gradient orientation and the strength of the derivative in any prescribed direction, for instance, the mean gradient direction. While this framework generalizes to 3D uncertain scalar fields, we concentrate on the visualization of the resulting intervals in 2D fields. We propose a novel color diffusion scheme to visualize both the absolute variability of the derivative strength and its magnitude relative to the mean values. A special family of circular glyphs is introduced to convey the uncertainty in gradient orientation. For a number of synthetic and real-world data sets, we demonstrate the use of our approach for analyzing the stability of certain features in uncertain 2D scalar fields, with respect to both local derivatives and feature orientation. In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings. The following statistical properties are derived for a random, moving, Gaussian surface: (1) the probability distribution of the surface elevation and of the magnitude and orientation of the gradient; (2) the average number of zero-crossings per unit distance along a line in an arbitrary direction; (3) the average length of the contours per unit area, and the distribution of their direction; (4) the average density of maxima and minima per unit area of the surface, and the average density of specular points (i.e, points where the two components of gradient take given values); (5) the probability distribution of the velocities of zero-crossings along a given line; (6) the probability distribution of the velocities of contours and of specular points; (7) the probability distribution of the envelope and phase angle, and hence (8) when the spectrum is narrow, the probability distribution of the heights of maxima and minima and the distribution of the intervals between successive zero-crossings along an arbitrary line. All the results are expressed in terms of the two-dimensional energy spectrum of the surface, and are found to involve the moments of the spectrum up to a finite order only. (1), (3), (4), (5) and (6) are discussed in detail for the special case of a narrow spectrum. The converse problem is also studied and solved: given certain statistical properties of the surface, to find a convergent sequence of approximations to the energy spectrum. The problems arise in connexion with the statistical analysis of the sea surface. (More detailed summaries are given at the beginning of each part of the paper.) We develop Riemannian Stein Variational Gradient Descent (RSVGD), a Bayesian inference method that generalizes Stein Variational Gradient Descent (SVGD) to Riemann manifold. The benefits are two-folds: (i) for inference tasks in Euclidean spaces, RSVGD has the advantage over SVGD of utilizing information geometry, and (ii) for inference tasks on Riemann manifolds, RSVGD brings the unique advantages of SVGD to the Riemannian world. To appropriately transfer to Riemann manifolds, we conceive novel and non-trivial techniques for RSVGD, which are required by the intrinsically different characteristics of general Riemann manifolds from Euclidean spaces. We also discover Riemannian Stein's Identity and Riemannian Kernelized Stein Discrepancy. Experimental results show the advantages over SVGD of exploring distribution geometry and the advantages of particle-efficiency, iteration-effectiveness and approximation flexibility over other inference methods on Riemann manifolds.