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.
In @cite_21 , the authors study (which are nothing more than maps @math except that @math and @math are surfaces, i.e. Riemannian manifolds of dimension @math ) and define a Dirichlet energy in the same way as Koorevaar, Schoen and Jost. These maps are seen as relaxations of "classical" maps @math , and they focus on numerical computation and visualization of theses soft maps, see also @cite_24 for applications to supervised learning. On the other hand, they do not analyze in detail the theoretical properties of the Dirichlet energy and harmonic mappings, which in contrast is the main topic of the present article. In @cite_32 , the author provides some theoretical analysis of soft maps by focusing on the cases where the boundary measures on @math are either Dirac masses or Gaussian measures.
@cite_24 @cite_21 @cite_32
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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 consider @math -equivariant wave maps from @math of finite energy. We establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map @math . In particular, for each global energy solution of topological degree one, we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term. Together with a companion article (Part I), where we consider the case of finite-time blow up, this gives a characterization of all @math -equivariant, degree @math wave maps in the energy regime @math . In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru (Commun. Math. Phys. 298(1):139–230, 2010; Commun. Math. Phys. 298(1):231–264, 2010), but more specifically into the unit Euclidean sphere ( S ^ n-1 R ^ n ) with ( n ), and study further the dynamics of the sequence of wave maps that are obtained in Sterbenz and Tataru (Commun. Math. Phys. 298(1):231–264, 2010) at the final rescaling for a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space. Similar and stronger results have been recently obtained in the equivariant setting by several authors (Cote, Commun. Pure Appl. Math. 68(11):1946–2004, 2015; Cote, Commun. Pure Appl. Math. 69(4):609–612, 2016; Cote, Am. J. Math. 137(1):139–207, 2015; , Am. J. Math. 137(1):209–250, 2015; Krieger, Commun. Math. Phys. 250(3):507–580, 2004), where better control on the dispersive term concentrating on the null boundary of the cone is provided, and in some cases the asymptotic decomposition is shown to hold for all time. Here, however, we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to Lin and Riviere (Ann. Math. 149(2):785–829, 1999; Duke Math. J. 111:177–193, 2002), which we make work here in the hyperbolic context of Sterbenz and Tataru (Commun. Math. Phys. 298(1), 231–264, 2010). Wave maps are the simplest wave equations taking their values in a Riemannian manifold (M,g). Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric g. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as @math , present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman–Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for @math as target. This monograph establishes that for @math as target the wave map evolution of any smooth data exists globally as a smooth function. While we restrict ourselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations. Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even infinite. Thus, the development of numerical methods for these parametric problems is faced with the possible curse of dimensionality. This article is directed at (i) identifying and understanding which properties of parametric equations allow one to avoid this curse and (ii) developing and analyzing effective numerical methodd which fully exploit these properties and, in turn, are immune to the growth in dimensionality. The first part of this article studies the smoothness and approximability of the solution map, that is, the map @math where @math is the parameter value and @math is the corresponding solution to the PDE. It is shown that for many relevant parametric PDEs, the parametric smoothness of this map is typically holomorphic and also highly anisotropic in that the relevant parameters are of widely varying importance in describing the solution. These two properties are then exploited to establish convergence rates of @math -term approximations to the solution map for which each term is separable in the parametric and physical variables. These results reveal that, at least on a theoretical level, the solution map can be well approximated by discretizations of moderate complexity, thereby showing how the curse of dimensionality is broken. This theoretical analysis is carried out through concepts of approximation theory such as best @math -term approximation, sparsity, and @math -widths. These notions determine a priori the best possible performance of numerical methods and thus serve as a benchmark for concrete algorithms. The second part of this article turns to the development of numerical algorithms based on the theoretically established sparse separable approximations. The numerical methods studied fall into two general categories. The first uses polynomial expansions in terms of the parameters to approximate the solution map. The second one searches for suitable low dimensional spaces for simultaneously approximating all members of the parametric family. The numerical implementation of these approaches is carried out through adaptive and greedy algorithms. An a priori analysis of the performance of these algorithms establishes how well they meet the theoretical benchmarks. In unsupervised domain mapping, the learner is given two unmatched datasets @math and @math . The goal is to learn a mapping @math that translates a sample in @math to the analog sample in @math . Recent approaches have shown that when learning simultaneously both @math and the inverse mapping @math , convincing mappings are obtained. In this work, we present a method of learning @math without learning @math . This is done by learning a mapping that maintains the distance between a pair of samples. Moreover, good mappings are obtained, even by maintaining the distance between different parts of the same sample before and after mapping. We present experimental results that the new method not only allows for one sided mapping learning, but also leads to preferable numerical results over the existing circularity-based constraint. Our entire code is made publicly available at this https URL . In this paper we study the topological properties of wireless communication maps and their usability in algorithmic design. We consider the SINR model, which compares the received power of a signal at a receiver against the sum of strengths of other interfering signals plus background noise. To describe the behavior of a multi-station network, we use the convenient representation of a . In the SINR model, the resulting partitions the plane into reception zones, one per station, and the complementary region of the plane where no station can be heard. We consider the general case where transmission energies are arbitrary (or non-uniform). Under that setting, the reception zones are not necessarily convex or even connected. This poses the algorithmic challenge of designing efficient point location techniques as well as the theoretical challenge of understanding the geometry of SINR diagrams. We achieve several results in both directions. We establish a form of weaker convexity in the case where stations are aligned on a line. In addition, one of our key results concerns the behavior of a @math -dimensional map. Specifically, although the @math -dimensional map might be highly fractured, drawing the map in one dimension higher "heals" the zones, which become connected. In addition, as a step toward establishing a weaker form of convexity for the @math -dimensional map, we study the interference function and show that it satisfies the maximum principle. Finally, we turn to consider algorithmic applications, and propose a new variant of approximate point location. To model modern large-scale datasets, we need efficient algorithms to infer a set of @math unknown model parameters from @math noisy measurements. What are fundamental limits on the accuracy of parameter inference, given finite signal-to-noise ratios, limited measurements, prior information, and computational tractability requirements? How can we combine prior information with measurements to achieve these limits? Classical statistics gives incisive answers to these questions as the measurement density @math . However, these classical results are not relevant to modern high-dimensional inference problems, which instead occur at finite @math . We formulate and analyze high-dimensional inference as a problem in the statistical physics of quenched disorder. Our analysis uncovers fundamental limits on the accuracy of inference in high dimensions, and reveals that widely cherished inference algorithms like maximum likelihood (ML) and maximum-a posteriori (MAP) inference cannot achieve these limits. We further find optimal, computationally tractable algorithms that can achieve these limits. Intriguingly, in high dimensions, these optimal algorithms become computationally simpler than MAP and ML, while still outperforming them. For example, such optimal algorithms can lead to as much as a 20 reduction in the amount of data to achieve the same performance relative to MAP. Moreover, our analysis reveals simple relations between optimal high dimensional inference and low dimensional scalar Bayesian inference, insights into the nature of generalization and predictive power in high dimensions, information theoretic limits on compressed sensing, phase transitions in quadratic inference, and connections to central mathematical objects in convex optimization theory and random matrix theory. The amoeba of a holomorphic function @math is, by definition, the image in @math of the zero locus of @math under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function @math with the property that its Monge-Ampere mass is concentrated to the amoeba of @math We obtain results of two kinds; by approximating @math with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of @math ; by computing the Monge-Ampere measure, we find sharp bounds for the area of amoebas in @math . We also consider systems of functions @math and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations. This paper uses probability theory to provide an approach to the Dirichlet problem for harmonic maps. The probabilistic tools used are those of manifold-valued Brownian motion and Γ-martingales. Probabilistic proofs are given of uniqueness, continuous dependence on data, and existence of generalized solutions (finely harmonic maps), for target domains which have convex geometry. (This class of domain includes all regular geodesic balls). Links are made with new results in other fields: the Γ-martingale Dirichlet problem, Riemannian centres of mass We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . The purpose of this paper is the detection of salient areas in natural video by using the new deep learning techniques. Salient patches in video frames are predicted first. Then the predicted visual fixation maps are built upon them. We design the deep architecture on the basis of CaffeNet implemented with Caffe toolkit. We show that changing the way of data selection for optimisation of network parameters, we can save computation cost up to @math times. We extend deep learning approaches for saliency prediction in still images with RGB values to specificity of video using the sensitivity of the human visual system to residual motion. Furthermore, we complete primary colour pixel values by contrast features proposed in classical visual attention prediction models. The experiments are conducted on two publicly available datasets. The first is IRCCYN video database containing @math videos with an overall amount of @math frames and eye fixations of @math subjects. The second one is HOLLYWOOD2 provided @math movie clips with the eye fixations of @math subjects. On IRCYYN dataset, the accuracy obtained is of @math . On HOLLYWOOD2 dataset, results in prediction of saliency of patches show the improvement up to @math with regard to RGB use only. The resulting accuracy of @math is obtained. The AUC metric in comparison of predicted saliency maps with visual fixation maps shows the increase up to @math on a sample of video clips from this dataset. We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points appear naturally in non-convex polynomial optimization which occurs in a wide range of scientific applications (control theory, chemistry, economics,...). Critical points also play a central role in recent algorithms of effective real algebraic geometry. Experimentally, it has been observed that Gr "obner basis algorithms are efficient to compute such points. Therefore, recent software based on the so-called Critical Point Method are built on Gr "obner bases engines. Let @math be polynomials in @math of degree @math , @math be their complex variety and @math be the projection map @math . The critical points of the restriction of @math to @math are defined by the vanishing of @math and some maximal minors of the Jacobian matrix associated to @math . Such a system is algebraically structured: the ideal it generates is the sum of a determinantal ideal and the ideal generated by @math . We provide the first complexity estimates on the computation of Gr "obner bases of such systems defining critical points. We prove that under genericity assumptions on @math , the complexity is polynomial in the generic number of critical points, i.e. @math . More particularly, in the quadratic case D=2, the complexity of such a Gr "obner basis computation is polynomial in the number of variables @math and exponential in @math . We also give experimental evidence supporting these theoretical results. Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called of a cover of the domain. In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage. More specifically, we show that the one-dimensional homology of the nerve complex @math of a path-connected cover @math of a domain @math cannot be richer than that of the domain @math itself. Intuitively, this result means that no new @math -homology class can be "created" under a natural map from @math to the nerve complex @math . Equipping @math with a pseudometric @math , we further refine this result and characterize the classes of @math that may survive in the nerve complex using the notion of of the covering elements in @math . These fundamental results about nerve complexes then lead to an analysis of the @math -homology of Reeb spaces, mappers and multiscale mappers. The analysis of @math -homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces. Let @math be a compact Riemannian submanifold of @math of dimension @math and let @math be a sample of i.i.d. points in @math with uniform distribution. We study the random operators @math where @math is the Gaussian kernel and @math as @math Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of @math @math (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations @math for smooth functions @math both pointwise and uniformly in @math and @math (here @math and @math is the Riemannian volume measure). In particular, we show that for any class @math of three times differentiable functions on @math with uniformly bounded derivatives @math as soon as @math and also prove asymptotic normality of @math (functional CLT) for a fixed @math and uniformly in @math We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the @math -dimensional lattice @math with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with parameter @math ) on some box of finite size @math . In this article, we replace the regularized corrector (which is the solution of a problem posed on @math ) by some practically computable proxy on some box of size @math , and quantify the associated additional error. In order to improve the convergence, one may also consider @math independent realizations of the computable proxy, and take the arithmetic mean of the associated approximate homogenized coefficients. A natural optimization problem consists in properly choosing @math and @math in order to reduce the error at given computational complexity. Our analysis is sharp and allows us to give a clear answer to this question. In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errors we derive. The efficiency of the approach is illustrated by a numerical study in dimension 2. We study identifiability for bilinear inverse problems under sparsity and subspace constraints. We show that, up to a global scaling ambiguity, almost all such maps are injective on the set of pairs of sparse vectors if the number of measurements @math exceeds @math , where @math and @math denote the sparsity of the two input vectors, and injective on the set of pairs of vectors lying in known subspaces of dimensions @math and @math if @math . We also prove that both these bounds are tight in the sense that one cannot have injectivity for a smaller number of measurements. Our proof technique draws from algebraic geometry. As an application we derive optimal identifiability conditions for the deconvolution problem, thus improving on recent work of [1]. For any integers @math and @math , we show the existence of a set of @math vectors @math such that any embedding @math satisfying @math must have @math This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of @math of interest, since there is always an isometric embedding into dimension @math (either the identity map, or projection onto @math ). Previously such a lower bound was only known to hold against linear maps @math , and not for such a wide range of parameters @math [LN16]. The best previously known lower bound for general @math was @math [Wel74, Lev83, Alo03], which is suboptimal for any @math . We consider the problem of finding the maximum area parallelogram (MAP) inside a given convex polygon. Our main result is an algorithm for computing the MAP in an @math -sided polygon in @math time. Achieving this running time requires proving several new structural properties of the MAP. Our algorithm actually computes all the locally maximal area parallelograms (LMAPs). In addition to the algorithm, we prove that the LMAPs interleave each other, thus the number of LMAPs is bounded by @math . We discuss applications of our result to, among others, the problem of computing the maximum area centrally-symmetric convex body (MAC) inside a convex polygon, and the simplest case of the Heilbronn Triangle Problem. Objective: To develop methods that automatically map abbreviations to their full forms in biomedical articles. @PARASPLIT Methods: The authors developed two methods of mapping defined and undefined abbreviations (defined abbreviations are paired with their full forms in the articles, whereas undefined ones are not). For defined abbreviations, they developed a set of pattern-matching rules to map an abbreviation to its full form and implemented the rules into a software program, AbbRE (for “abbreviation recognition and extraction”). Using the opinions of domain experts as a reference standard, they evaluated the recall and precision of AbbRE for defined abbreviations in ten biomedical articles randomly selected from the ten most frequently cited medical and biological journals. They also measured the percentage of undefined abbreviations in the same set of articles, and they investigated whether they could map undefined abbreviations to any of four public abbreviation databases (GenBank LocusLink, swissprot, LRABR of the UMLS Specialist Lexicon, and Bioabacus). @PARASPLIT Results: AbbRE had an average 0.70 recall and 0.95 precision for the defined abbreviations. The authors found that an average of 25 percent of abbreviations were defined in biomedical articles and that of a randomly selected subset of undefined abbreviations, 68 percent could be mapped to any of four abbreviation databases. They also found that many abbreviations are ambiguous (i.e., they map to more than one full form in abbreviation databases). @PARASPLIT Conclusion: AbbRE is efficient for mapping defined abbreviations. To couple AbbRE with abbreviation databases for the mapping of undefined abbreviations, not only exhaustive abbreviation databases but also a method to resolve the ambiguity of abbreviations in the databases are needed.