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Simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing and is typically only feasible by using approximate Markov chain Monte Carlo sampling. We propose a minimax tilting method for exact independently and identically distributed data simulation from the truncated multivariate normal distribution. The new methodology provides both a method for simulation and an efficient estimator to hitherto intractable Gaussian integrals. We prove that the estimator has a rare vanishing relative error asymptotic property. Numerical experiments suggest that the scheme proposed is accurate in a wide range of set-ups for which competing estimation schemes fail. We give an application to exact independently and identically distributed data simulation from the Bayesian posterior of the probit regression model.

We consider a class of nonlinear mappings FA,N in ℝN indexed by symmetric random matrices A ∈ ℝN×N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333–366]. Within information theory, they are known as \"approximate message passing\" algorithms. We study the high-dimensional (large N) behavior of the iterates of F for polynomial functions F, and prove that it is universal; that is, it depends only on the first two moments of the entries of A, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.

The Gaussian polytope $\\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\\mathbb{R}^d$ . We derive explicit expressions for the probability that $\\mathcal P_{n,d}$ contains a fixed point $x\\in\\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\\mathcal P_{n,d}$ contains the point $\\sigma X$ , where $\\sigma\\geq 0$ is constant and X is a standard normal vector independent of $\\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\\Phi(z)$ and its complex version $\\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

We study maximum likelihood estimation for the statistical model for undirected random graphs, known as the β-model, in which the degree sequences are minimal sufficient statistics. We derive necessary and sufficient conditions, based on the polytope of degree sequences, for the existence of the maximum likelihood estimator (MLE) of the model parameters. We characterize in a combinatorial fashion sample points leading to a nonexistent MLE, and nonestimability of the probability parameters under a nonexistent MLE. We formulate conditions that guarantee that the MLE exists with probability tending to one as the number of nodes increases.

Let \\[U_1,U_2,\\ldots \\] be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as \\[n\\rightarrow \\infty \\], the f-vector of the \\[(d+1)\\]-dimensional convex cone \\[C_n\\] generated by \\[U_1,\\ldots ,U_n\\] weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f-vector of \\[C_n\\] and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of \\[C_n\\] can be expressed through the expected f-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Bárány et al. (Random Struct Algorithms 50(1):3–22, 2017. https://doi.org/10.1002/rsa.20644). Our approach is based on the observation that the random cone \\[C_n\\] weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to \\[\\Vert x\\Vert ^{-(d+\\gamma )}\\], where \\[\\gamma =1\\]. We compute the expected number of facets, the expected intrinsic volumes and the expected T-functional of this random convex hull for arbitrary \\[\\gamma >0\\].

Graphical models have attracted increasing attention in recent years, especially in settings involving high-dimensional data. In particular, Gaussian graphical models are used to model the conditional dependence structure among multiple Gaussian random variables. As a result of its computational efficiency, the graphical lasso (glasso) has become one of the most popular approaches for fitting high-dimensional graphical models. In this paper, we extend the graphical models concept to model the conditional dependence structure among p random functions. In this setting, not only is p large, but each function is itself a high-dimensional object, posing an additional level of statistical and computational complexity. We develop an extension of the glasso criterion (fglasso), which estimates the functional graphical model by imposing a block sparsity constraint on the precision matrix, via a group lasso penalty. The fglasso criterion can be optimized using an efficient block coordinate descent algorithm. We establish the concentration inequalities of the estimates, which guarantee the desirable graph support recovery property, that is, with probability tending to one, the fglasso will correctly identify the true conditional dependence structure. Finally, we show that the fglasso significantly outperforms possible competing methods through both simulations and an analysis of a real-world electroencephalography dataset comparing alcoholic and nonalcoholic patients.

Given a probability measure μ on Rn, Tukey’s half-space depth is defined for any x∈Rn by φμ(x)=inf{μ(H):H∈H(x)}, where H(x) is the set of all half-spaces H of Rn containing x. We show that if μ is a non-degenerate log-concave probability measure on Rn then e-c1n⩽∫Rnφμ(x)dμ(x)⩽e-c2n/Lμ2where Lμ is the isotropic constant of μ and c1,c2>0 are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of Lq-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.

Time-varying polytope distance (TVPD) problems are prevalent in scientific and engineering applications and can be transformed into time-varying quadratic programming (TVQP) problems with both equality and inequality constraints. Concurrently, the noise interferences during the solution process are non-negligible and challenging to eliminate. Although zeroing neural networks (ZNNs) perform well in solving various types of time-varying problems, they still fall short in the suppression of unbounded noises, such as linear noise. To address this limitation, this paper proposes an improving integration-enhanced ZNN (IIEZNN) model for accurately solving TVPD problems under noise environments. Compared with the existing ZNN models, the IIEZNN model has stronger inherent robustness. The stability and robustness of the IIEZNN model are guaranteed by rigorous theoretical analysis. Firstly, the effectiveness of the IIEZNN model is verified via two TVQP examples. Then, the IIEZNN model is generalized to TVPD problem solving and has excellent performance. Specifically, in solving the TVPD under linear noises, the residual error of the IIEZNN model converges to the order of 10 - 5 , which is much lower than that of the existing noise-tolerant ZNN model with an order of 10 - 1 .

We establish power-series expansions for the asymptotic expectations of the vertex number and missed area of random disc-polygons in planar convex bodies with $C^{k+1}_+$ -smooth boundaries. These results extend asymptotic formulas proved in Fodor et al. (2014).

The β-model of random graphs is an exponential family model with the degree sequence as a sufficient statistic. In this paper, we contribute three key results. First, we characterize conditions that lead to a quadratic time algorithm to check for the existence of MLE of the β-model, and show that the MLE never exists for the degree partition β-model. Second, motivated by privacy problems with network data, we derive a differentially private estimator of the parameters of β-model, and show it is consistent and asymptotically normally distributed—it achieves the same rate of convergence as the nonprivate estimator. We present an efficient algorithm for the private estimator that can be used to release synthetic graphs. Our techniques can also be used to release degree distributions and degree partitions accurately and privately, and to perform inference from noisy degrees arising from contexts other than privacy. We evaluate the proposed estimator on real graphs and compare it with a current algorithm for releasing degree distributions and find that it does significantly better. Finally, our paper addresses shortcomings of current approaches to a fundamental problem of how to perform valid statistical inference from data released by privacy mechanisms, and lays a foundational groundwork on how to achieve optimal and private statistical inference in a principled manner by modeling the privacy mechanism; these principles should be applicable to a class of models beyond the β-model.