Given a set @math of @math points in the plane, we study the computation of the probability distribution function of both the area and perimeter of the convex hull of a random subset @math of @math . The random subset @math is formed by drawing each point @math of @math independently with a given rational probability @math . For both measures of the convex hull, we show that it is #P-hard to compute the probability that the measure is at least a given bound @math . For @math , we provide an algorithm that runs in @math time and returns a value that is between the probability that the area is at least @math , and the probability that the area is at least @math . For the perimeter, we show a similar algorithm running in @math time. Finally, given @math and for any measure, we show an @math -time Monte Carlo algorithm that returns a value that, with probability of success at least @math , differs at most @math from the probability that the measure is at least @math .
Stochastic finite point sets in the plane, as the one considered in this paper, appear in a natural manner in many database scenarios in which the gathered data has many false positives @cite_2 @cite_16 @cite_10 . This model of random points differs from the model in which @math points are chosen independently at random in some Euclidean region, and questions related to the final positions of the points are considered @cite_13 @cite_5 @cite_1 .
@cite_10 @cite_1 @cite_2 @cite_5 @cite_16 @cite_13
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The binary symmetric stochastic block model deals with a random graph of @math vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability @math within clusters and @math across clusters. In the asymptotic regime of @math and @math for fixed @math and @math , we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe14 . Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to @math . We consider the stochastic geometry model where the location of each node is a random point in a given metric space, or the existence of each node is uncertain. We study the problems of computing the expected lengths of several combinatorial or geometric optimization problems over stochastic points, including closest pair, minimum spanning tree, (k )-clustering, minimum perfect matching, and minimum cycle cover. We also consider the problem of estimating the probability that the length of closest pair, or the diameter, is at most, or at least, a given threshold. Most of the above problems are known to be ( # P )-hard. We obtain FPRAS (Fully Polynomial Randomized Approximation Scheme) for most of them in both the existential and locational uncertainty models. Our result for stochastic minimum spanning trees in the locational uncertain model improves upon the previously known constant factor approximation algorithm. Our results for other problems are the first known to the best of our knowledge. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph @math from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and --in particular-- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. This paper deals with a new class of random flights in ℝ d , d≥2, characterized by non-uniform probability distributions on the multidimensional sphere. These random motions differ from similar models appeared in literature where the directions are taken according to the uniform law. The family of angular probability distributions introduced in this paper depends on a parameter ν≥0, which gives the anisotropy of the motion. Furthermore, we assume that the number of changes of direction performed by the random flight is fixed. The time lengths between two consecutive changes of orientation have joint probability distribution given by a Dirichlet density function. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. Stochastic geometry proves to be a powerful tool for modeling dense wireless networks adopting random MAC protocols such as ALOHA and CSMA. The main strength of this methodology lies in its ability to account for the randomness in the nodes' location jointly with an accurate description at the physical layer, based on the SINR, that allows to consider also random fading on each link. Existing models of CSMA networks adopting the stochastic geometry approach suffer from two important weaknesses: 1) they permit to evaluate only spatial averages of the main performance measures, thus hiding possibly huge discrepancies in the performance achieved by individual nodes; 2) they are analytically tractable only when nodes are distributed over the area according to simple spatial processes (e.g., the Poisson point process). In this paper we show how the stochastic geometry approach can be extended to overcome the above limitations, allowing to obtain node throughput distributions as well as to analyze a significant class of topologies in which nodes are not independently placed. With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over such data. In this paper, we initiate the study of constructing epsilon-kernel coresets for uncertain points. We consider uncertainty in the existential model where each point's location is fixed but only occurs with a certain probability, and the locational model where each point has a probability distribution describing its location. An epsilon-kernel coreset approximates the width of a point set in any direction. We consider approximating the expected width (an epsilon-EXP-KERNEL), as well as the probability distribution on the width (an (epsilon, tau)-QUANT-KERNEL) for any direction. We show that there exists a set of O(epsilon^ -(d-1) 2 ) deterministic points which approximate the expected width under the existential and locational models, and we provide efficient algorithms for constructing such coresets. We show, however, it is not always possible to find a subset of the original uncertain points which provides such an approximation. However, if the existential probability of each point is lower bounded by a constant, an epsilon-EXP-KERNEL is still possible. We also provide efficient algorithms for construct an (epsilon, tau)-QUANT-KERNEL coreset in nearly linear time. Our techniques utilize or connect to several important notions in probability and geometry, such as Kolmogorov distances, VC uniform convergence and Tukey depth, and may be useful in other geometric optimization problem in stochastic settings. Finally, combining with known techniques, we show a few applications to approximating the extent of uncertain functions, maintaining extent measures for stochastic moving points and some shape fitting problems under uncertainty. We consider the problem of community detection in the Stochastic Block Model with a finite number @math of communities of sizes linearly growing with the network size @math . This model consists in a random graph such that each pair of vertices is connected independently with probability @math within communities and @math across communities. One observes a realization of this random graph, and the objective is to reconstruct the communities from this observation. We show that under spectral algorithms, the number of misclassified vertices does not exceed @math with high probability as @math grows large, whenever @math , @math and where @math and @math denote the (fixed) proportions of vertices in the two smallest communities. In view of recent work by and , this establishes that the proposed spectral algorithms are able to exactly recover communities whenever this is at all possible in the case of networks with two communities with equal sizes. We conjecture that condition (1) is actually necessary to obtain less than @math misclassified vertices asymptotically, which would establish the optimality of spectral method in more general scenarios. We study the complexity of geometric minimum spanning trees under a stochastic model of input: Suppose we are given a master set of points s1,s_2,...,sn in d-dimensional Euclidean space, where each point si is active with some independent and arbitrary but known probability pi. We want to compute the expected length of the minimum spanning tree (MST) of the active points. This particular form of stochastic problems is motivated by the uncertainty inherent in many sources of geometric data but has not been investigated before in computational geometry to the best of our knowledge. Our main results include the following. We show that the stochastic MST problem is SPHARD for any dimension d ≥ 2. We present a simple fully polynomial randomized approximation scheme (FPRAS) for a metric space, and thus also for any Euclidean space. For d=2, we present two deterministic approximation algorithms: an O(n4)-time constant-factor algorithm, and a PTAS based on a combination of shifted quadtrees and dynamic programming. We show that in a general metric space the tail bounds of the distribution of the MST length cannot be approximated to any multiplicative factor in polynomial time under the assumption that P ≠ NP. In addition to this existential model of stochastic input, we also briefly consider a locational model where each point is present with certainty but its location is probabilistic. Stochastic geometry models for wireless communication networks have recently attracted much attention. This is because the performance of such networks critically depends on the spatial configuration of wireless nodes and the irregularity of the node configuration in a real network can be captured by a spatial point process. However, most analysis of such stochastic geometry models for wireless networks assumes, owing to its tractability, that the wireless nodes are deployed according to homogeneous Poisson point processes. This means that the wireless nodes are located independently of each other and their spatial correlation is ignored. In this work we propose a stochastic geometry model of cellular networks such that the wireless base stations are deployed according to the Ginibre point process. The Ginibre point process is one of the determinantal point processes and accounts for the repulsion between the base stations. For the proposed model, we derive a computable representation for the coverage probability—the probability that the signal-to-interference-plus-noise ratio (SINR) for a mobile user achieves a target threshold. To capture its qualitative property, we further investigate the asymptotics of the coverage probability as the SINR threshold becomes large in a special case. We also present the results of some numerical experiments. 1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random high-dimensional polytopes; a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications - in statistics, probability, information theory, and signal processing - with potential impacts in practical subjects like medical imaging and digital communications. Before involving the reader in our lengthy analysis of high-dimensional face counting, we describe three implications of our results. 1.1.1. Convex Hulls of Gaussian Point Clouds. Consider a random point cloud of n points xi, i = 1, . . . , n, sampled independently and identically from a Gaussian distribution in R d with nonsingular covariance. This is a standard model of multivariate data; its properties are increasingly important in a wide range of applications. At the same time, it is an attractive and in some sense timeless object for theoretical study. Properties of the convex hull of the random point cloud X = xi have attracted interest for several decades, increasingly so in recent years; there is a nowvoluminous literature on the subject. The results could be significant for understanding outlier detection, or classification problems in machine learning. We study the asymptotic statistical behavior of the 2-dimensional periodic Lorentz gas with an infinite horizon. We consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers. We assume that the initial position of the particle in the phase space is random with uniform distribution with respect to the Liouville measure of the periodic problem. We are interested in the asymptotic statistical behavior of the particle displacement in the plane as the timet goes to infinity. We assume that the particle horizon is infinite, which means that the length of free motion of the particle is unbounded. Then we show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement in the plane is Gaussian, but the normalization factor is (t logt)1 2 and nott1 2 as in the classical case. We find the covariance matrix of the limit distribution. The stochastic block model (SBM) with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behaviour. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability @math within clusters and @math across clusters. In the past two decades, a large body of literature in statistics and computer science has focused on providing lower-bounds on the scaling of @math to ensure exact recovery. In this paper, we identify a sharp threshold phenomenon for exact recovery: if @math and @math are constant (with @math ), recovering the communities with high probability is possible if @math and impossible if @math . In particular, this improves the existing bounds. This also sets a new line of sight for efficient clustering algorithms. While maximum likelihood (ML) achieves the optimal threshold (by definition), it is in the worst-case NP-hard. This paper proposes an efficient algorithm based on a semidefinite programming relaxation of ML, which is proved to succeed in recovering the communities close to the threshold, while numerical experiments suggest it may achieve the threshold. An efficient algorithm which succeeds all the way down to the threshold is also obtained using a partial recovery algorithm combined with a local improvement procedure. The labeled stochastic block model is a random graph model representing networks with community structure and interactions of multiple types. In its simplest form, it consists of two communities of approximately equal size, and the edges are drawn and labeled at random with probability depending on whether their two endpoints belong to the same community or not. It has been conjectured in [1] that this model exhibits a phase transition: reconstruction (i.e. identification of a partition positively correlated with the “true partition” into the underlying communities) would be feasible if and only if a model parameter exceeds a threshold. We prove one half of this conjecture, i.e., reconstruction is impossible when below the threshold. In the converse direction, we introduce a suitably weighted graph. We show that when above the threshold by a specific constant, reconstruction is achieved by (1) minimum bisection, and (2) a spectral method combined with removal of nodes of high degree. We present a novel probabilistic approach for fitting a statistical model to an image. A 3D Morphable Model (3DMM) of faces is interpreted as a generative (Top-Down) Bayesian model. Random Forests are used as noisy detectors (Bottom-Up) for the face and facial landmark positions. The Top-Down and Bottom-Up parts are then combined using a Data-Driven Markov Chain Monte Carlo Method (DDMCMC). As core of the integration, we use the Metropolis-Hastings algorithm which has two main advantages. First, the algorithm can handle unreliable detections and therefore does not need the detectors to take an early and possible wrong hard decision before fitting. Second, it is open for integration of various cues to guide the fitting process. Based on the proposed approach, we implemented a completely automatic, pose and illumination invariant face recognition application. We are able to train and test the building blocks of our application on different databases. The system is evaluated on the Multi-PIE database and reaches state of the art performance. [1] conjectured the existence of a sharp threshold on model parameters for community detection in sparse random graphs drawn from the stochastic block model. Mossel, Neeman and Sly [2] established the negative part of the conjecture, proving impossibility of non-trivial reconstruction below the threshold. In this work we solve the positive part of the conjecture. To that end we introduce a modified adjacency matrix B which counts self-avoiding paths of a given length e between pairs of nodes. We then prove that for logarithmic length e, the leading eigenvectors of this modified matrix provide a non-trivial reconstruction of the underlying structure, thereby settling the conjecture. A key step in the proof consists in establishing a weak Ramanujan property of the constructed matrix B. Namely, the spectrum of B consists in two leading eigenvalues ρ(B), λ2 and n -- 2 eigenvalues of a lower order O(ne √ρ(B) for all e 0, ρ(B) denoting B's spectral radius. We introduce a model of random interlacements made of a countable collection of doubly infinite paths on ℤ d ,d > 3. A nonnegative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder (ℤ Nℤ) d-1 x ℤ by simple random walk, or the set of points visited by simple random walk on the discrete torus (ℤ Nℤ) d at times of order u N d . In particular we study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite connected translation invariant random subset of ℤ d . We introduce a critical value u * such that the vacant set percolates for u u * . Our main results show that u * is finite when d > 3 and strictly positive when d > 7. Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0,1]d, and connecting two points if their Euclidean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of @math points distributed uniformly in [0,1]d. We present upper bounds on the threshold width, and show that our bound is sharp for d = 1 and at most a sublogarithmic factor away for d ≥ 2. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs. 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. The stochastic block model is one of the oldest and most ubiquitous models for studying clustering and community detection. In an exciting sequence of developments, motivated by deep but non-rigorous ideas from statistical physics, conjectured a sharp threshold for when community detection is possible in the sparse regime. Mossel, Neeman and Sly and Massoulie proved the conjecture and gave matching algorithms and lower bounds. Here we revisit the stochastic block model from the perspective of semirandom models where we allow an adversary to make helpful' changes that strengthen ties within each community and break ties between them. We show a surprising result that these helpful' changes can shift the information-theoretic threshold, making the community detection problem strictly harder. We complement this by showing that an algorithm based on semidefinite programming (which was known to get close to the threshold) continues to work in the semirandom model (even for partial recovery). This suggests that algorithms based on semidefinite programming are robust in ways that any algorithm meeting the information-theoretic threshold cannot be. These results point to an interesting new direction: Can we find robust, semirandom analogues to some of the classical, average-case thresholds in statistics? We also explore this question in the broadcast tree model, and we show that the viewpoint of semirandom models can help explain why some algorithms are preferred to others in practice, in spite of the gaps in their statistical performance on random models.