We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graph-based machine learning algorithms for tasks such as clustering. In particular, we study when the cut capacity, and more generally total variation, on these graphs is a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of Γ-convergence. We obtain almost optimal conditions on the scaling, as the number of points increases, of the size of the neighborhood over which the points are connected by an edge for the Γ-convergence to hold. Taking of the limit is enabled by a transportation based metric which allows us to suitably compare functionals defined on different point clouds.
In the discrete setting, works related to the @math -convergence of functionals to continuous functionals involving perimeter include @cite_44 , @cite_55 and @cite_36 . The results by Braides and Yip @cite_44 , can be interpreted as the analogous results in a discrete setting to the ones obtained by Modica and Mortola. They give the description of the limiting functional (in the sense of @math -convergence) after appropriately rescaling the energies. In the discretized version considered, they work on a regular grid and the gradient term gets replaced by a finite-difference approximation that depends on the mesh size @math . Van Gennip and Bertozzi @cite_55 consider a similar problem and obtain analogous results. In @cite_36 , Chambolle, Giacomini and Lussardi consider a very general class of anisotropic perimeters defined on discrete subsets of a finite lattice of the form @math . They prove the @math -convergence of the functionals as @math to an anisotropic perimeter defined on a given domain in @math .
@cite_44 @cite_55 @cite_36
2086002908 2963833097 2129572061
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 . We investigate the limiting description for a finite-difference approximation of a singularly perturbed Allen--Cahn type energy functional. The key issue is to understand the interaction between two small length-scales: the interfacial thickness @math and the mesh size of spatial discretization @math . Depending on their relative sizes, we obtain results in the framework of @math -convergence for the (i) subcritical ( @math ), (ii) critical ( @math ), and (iii) supercritical ( @math ) cases. The first case leads to the same area functional as the spatially continuous case while the third gives the same result as that coming from a ferromagnetic spin energy. The critical case can be regarded as an interpolation between the two. This is the second article of a series of papers on stochastic homogenization of discrete elliptic equations. We consider a discrete elliptic equation on the @math -dimensional lattice @math with random coefficients @math of the simplest type: They are identically distributed and independent from edge to edge. On scales large w. r. t. the lattice spacing (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric ''homogenized'' matrix @math is characterized by @math for any direction @math , where the random field @math (the ''corrector'') is the unique solution of @math in @math such that @math , @math is stationary and @math , @math denoting the ensemble average (or expectation). In order to approximate the homogenized coefficients @math , the corrector problem is usually solved in a box @math of size @math with periodic boundary conditions, and the space averaged energy on @math defines an approximation @math of @math . Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation @math converges almost surely to @math as @math . In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size @math , but replace the elliptic operator by @math with (typically) @math , as standard in the homogenization literature. We then replace the ensemble average by a space average on @math , and estimate the overall error on the homogenized coefficients in terms of @math and @math . We investigate a family of regression problems in a semi-supervised setting. The task is to assign real-valued labels to a set of @math sample points, provided a small training subset of @math labeled points. A goal of semi-supervised 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 @math , 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 the discrete @math -Laplacian regularization. We investigate asymptotic behavior when the number of unlabeled points increases, while the number of training points remains fixed. We uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. We rigorously obtain almost optimal ranges on the scaling of @math for the asymptotic consistency to hold. We prove that the minimizers of the discrete functionals in random setting converge uniformly to the desired continuum limit. Furthermore we discover that for the standard model used there is a restrictive upper bound on how quickly @math must converge to zero as @math . We introduce a new model which is as simple as the original model, but overcomes this restriction. 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 consider the minimization of non-convex quadratic forms regularized by a cubic term, which exhibit multiple saddle points and poor local minima. Nonetheless, we prove that, under mild assumptions, gradient descent approximates the @math to within @math accuracy in @math steps for large @math and @math steps for small @math (compared to a condition number we define), with at most logarithmic dependence on the problem dimension. When we use gradient descent to approximate the Nesterov-Polyak cubic-regularized Newton step, our result implies a rate of convergence to second-order stationary points of general smooth non-convex functions. In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient ascent provide a factor @math approximation to the global maxima. We also study stochastic gradient methods and show that after @math iterations these methods reach solutions which achieve in expectation objective values exceeding @math . An immediate application of our results is to maximize submodular functions that are defined stochastically, i.e. the submodular function is defined as an expectation over a family of submodular functions with an unknown distribution. We will show how stochastic gradient methods are naturally well-suited for this setting, leading to a factor @math approximation when the function is monotone. In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains. Finally, experiments on real data demonstrate that our projected gradient methods consistently achieve the best utility compared to other continuous baselines while remaining competitive in terms of computational effort. We consider in this paper the problem of sampling a high-dimensional probability distribution @math having a density with respect to the Lebesgue measure on @math , known up to a normalization constant @math . Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that @math is continuously differentiable, @math is globally Lipschitz and @math is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order @math and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims. The Ginzburg-Landau functional is a phase transition model which is suitable for clustering or classification type problems. We study the asymptotics of a sequence of Ginzburg-Landau functionals with anisotropic interaction potentials on point clouds @math where @math denotes the number data points. In particular we show the limiting problem, in the sense of @math -convergence, is related to the total variation norm restricted to functions taking binary values; which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence. Abstract Irregular sampling and “stable sampling” of band-limited functions have been studied by H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52]. We prove that quasicrystals are sets of stable sampling. To cite this article: B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008). A discrete uncoupling process for finite spaces is introduced, called the Markov Cluster Process or the MCL process. The process is the engine for the graph clustering algorithm called the MCL algorithm. The MCL process takes a stochastic matrix as input, and then alternates expansion and inflation, each step defining a stochastic matrix in terms of the previous one. Expansion corresponds with taking the @math th power of a stochastic matrix, where @math . Inflation corresponds with a parametrized operator @math , @math , that maps the set of (column) stochastic matrices onto itself. The image @math is obtained by raising each entry in @math to the @math th power and rescaling each column to have sum 1 again. In practice the process converges very fast towards a limit that is invariant under both matrix multiplication and inflation, with quadratic convergence around the limit points. The heuristic behind the process is its expected behavior for (Markov) graphs possessing cluster structure. The process is typically applied to the matrix of random walks on a given graph @math , and the connected components of (the graph associated with) the process limit generically allow a clustering interpretation of @math . The limit is in general extremely sparse and iterands are sparse in a weighted sense, implying that the MCL algorithm is very fast and highly scalable. Several mathematical properties of the MCL process are established. Most notably, the process (and algorithm) iterands posses structural properties generalizing the mapping from process limits onto clusterings. The inflation operator @math maps the class of matrices that are diagonally similar to a symmetric matrix onto itself. The phrase diagonally positive semi-definite (dpsd) is used for matrices that are diagonally similar to a positive semi-definite matrix. For @math and for @math a stochastic dpsd matrix, the image @math is again dpsd. Determinantal inequalities satisfied by a dpsd matrix @math imply a natural ordering among the diagonal elements of @math , generalizing the mapping of process limits onto clusterings. The spectrum of @math is of the form @math , where @math is the number of endclasses of the ordering associated with @math , and @math is the dimension of @math . This attests to the uncoupling effect of the inflation operator. Sampling distribution over high-dimensional state-space is a problem which has recently attracted a lot of research efforts; applications include Bayesian non-parametrics, Bayesian inverse problems and aggregation of estimators. All these problems boil down to sample a target distribution @math having a density the Lebesgue measure on @math , known up to a normalisation factor @math where @math is continuously differentiable and smooth. In this paper, we study a sampling technique based on the Euler discretization of the Langevin stochastic differential equation. Contrary to the Metropolis Adjusted Langevin Algorithm (MALA), we do not apply a Metropolis-Hastings correction. We obtain for both constant and decreasing step sizes in the Euler discretization, non-asymptotic bounds for the convergence to stationarity in both total variation and Wasserstein distances. A particular attention is paid on the dependence on the dimension of the state space, to demonstrate the applicability of this method in the high dimensional setting, at least when @math is convex. These bounds are based on recently obtained estimates of the convergence of the Langevin diffusion to stationarity using Poincar 'e and log-Sobolev inequalities. These bounds improve and extend the results of (Dalalyan, 2014). We also investigate the convergence of an appropriately weighted empirical measure and we report sharp bounds for the mean square error and exponential deviation inequality for Lipschitz functions. A limited Monte Carlo experiment is carried out to support our findings. We consider a large class of random geometric graphs constructed from samples @math of independent, identically distributed observations of an underlying probability measure @math on a bounded domain @math . The popular modularity' clustering method specifies a partition @math of the set @math as the solution of an optimization problem. In this paper, under conditions on @math and @math , we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions @math converge in a certain sense to a continuum partition @math of the underlying domain @math , characterized as the solution of a type of Kelvin's shape optimization problem. Let @math denote a set of polynomials with complex coefficients. Let @math denote any set of sample points . For any @math , the discrete polynomial transform of @math (with respect to @math and @math ) is defined as the collection of sums, @math , where @math for some associated weight function @math . These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper, we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of @math orthogonal polynomials of degree at most @math , we give an @math algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the @math operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a three-term recurrence and thus it may be applied to any such set of discretely sampled functions. In particular, sampled orthogonal polynomials generate the vector space of functions on a distance transitive graph. As a direct application of our work, we are able to give a fast algorithm for computing subspace decompositions of this vector space which respect the action of the symmetry group of such a graph. This has direct applications to treating computational bottlenecks in the spectral analysis of data on distance transitive graphs, and we discuss this in some detail. The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding, and representable discrete polymatroids. We consider the vector linear solutions of networks over a field @math , with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. It is well known that a scalar linear solution over @math exists for a network if and only if the network is matroidal with respect to a matroid representable over @math . We define a discrete polymatroidal network and show that a linear fractional solution over a field @math exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over @math . An algorithm to construct the networks starting from certain class of discrete polymatroids is provided. Every representation over @math for the discrete polymatroid, results in a linear fractional solution over @math for the constructed network. Next, we consider the index coding problem, which involves a sender, which generates a set of messages @math , and a set of receivers @math , which demand messages. A receiver @math is specified by the tuple @math , where @math is the message demanded by @math and @math is the side information possessed by @math . We first show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid, satisfying certain conditions, which are determined by the index coding problem considered. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that matroid. The multi-linear representation of a matroid can be viewed as a special case of representation of an appropriate discrete polymatroid. We generalize the result of , by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid. Many data-fitting applications require the solution of an optimization problem involving a sum of large number of functions of high dimensional parameter. Here, we consider the problem of minimizing a sum of @math functions over a convex constraint set @math where both @math and @math are large. In such problems, sub-sampling as a way to reduce @math can offer great amount of computational efficiency. Within the context of second order methods, we first give quantitative local convergence results for variants of Newton's method where the Hessian is uniformly sub-sampled. Using random matrix concentration inequalities, one can sub-sample in a way that the curvature information is preserved. Using such sub-sampling strategy, we establish locally Q-linear and Q-superlinear convergence rates. We also give additional convergence results for when the sub-sampled Hessian is regularized by modifying its spectrum or Levenberg-type regularization. Finally, in addition to Hessian sub-sampling, we consider sub-sampling the gradient as way to further reduce the computational complexity per iteration. We use approximate matrix multiplication results from randomized numerical linear algebra (RandNLA) to obtain the proper sampling strategy and we establish locally R-linear convergence rates. In such a setting, we also show that a very aggressive sample size increase results in a R-superlinearly convergent algorithm. While the sample size depends on the condition number of the problem, our convergence rates are problem-independent, i.e., they do not depend on the quantities related to the problem. Hence, our analysis here can be used to complement the results of our basic framework from the companion paper, [38], by exploring algorithmic trade-offs that are important in practice. We consider a discrete elliptic equation with random coefficients @math , which (to fix ideas) are identically distributed and independent from grid point to grid point @math . On scales large w. r. t. the grid size (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients @math are characterized by @math where the random field @math is the unique stationary solution of the ''corrector problem'' @math and @math denotes the ensemble average. It is known (''by ergodicity'') that the above ensemble average of the energy density @math , which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of @math on length scales @math is estimated as follows: @math where the averaging function (i. e. @math , @math ) has to be smooth in the sense that @math . In two space dimensions (i. e. @math ), there is a logarithmic correction. In other words, smooth averages of the energy density @math behave like as if @math would be independent from grid point to grid point (which it is not for @math ). This result is of practical significance, since it allows to estimate the error when numerically computing @math . We provide an O(log log opt)-approximation algorithm for the problem of guarding a simple polygon with guards on the perimeter. We first design a polynomial-time algorithm for building e-nets of size @math for the instances of Hitting Set associated with our guarding problem. We then apply the technique of Bronnimann and Goodrich to build an approximation algorithm from this e-net finder. Along with a simple polygon P, our algorithm takes as input a finite set of potential guard locations that must include the polygon’s vertices. If a finite set of potential guard locations is not specified, e.g., when guards may be placed anywhere on the perimeter, we use a known discretization technique at the cost of making the algorithm’s running time potentially linear in the ratio between the longest and shortest distances between vertices. Our algorithm is the first to improve upon O(log opt)-approximation algorithms that use generic net finders for set systems of finite VC-dimension. Given a discrete random walk on a finite graph @math , the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step @math . These sets induce subgraphs of the underlying graph. Let @math be the subgraph of @math induced by the vacant set of the walk at step @math . Similarly, let @math be the subgraph of @math induced by the edges of the vacant net. For random @math -regular graphs @math , it was previously established that for a simple random walk, the graph @math of the vacant set undergoes a phase transition in the sense of the phase transition on Erd os -Renyi graphs @math . Thus, for @math there is an explicit value @math of the walk, such that for @math , @math has a unique giant component, plus components of size @math , whereas for @math all the components of @math are of size @math . We establish the threshold value @math for a phase transition in the graph @math of the vacant net of a simple random walk on a random @math -regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for @math even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random @math -regular graphs. The main findings are the following: As @math increases the threshold for the vacant set converges to @math in all three walks. For the vacant net, the threshold converges to @math for both the simple random walk and non-backtracking random walk. When @math is even, the threshold for the vacant net of the unvisited edge process converges to @math , which is also the vertex cover time of the process.