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We consider a multivariate piecewise linear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured by the integrated mean square error. Multivariate piecewise linear interpolator is…
Let $X= \{X(p), p\in M\}$ be a centered Gaussian random field, where $M$ is a smooth Riemannian manifold. For a suitable compact subset $D\subset M$, we obtain the approximations to excursion probability $\mathbb{P}\{\sup_{p\in D} X(p) \ge…
Entity Set Expansion is an important NLP task that aims at expanding a small set of entities into a larger one with items from a large pool of candidates. In this paper, we propose GausSetExpander, an unsupervised approach based on optimal…
We develop a theory of optimal transport for stationary random measures with a focus on stationary point processes and construct a family of distances on the set of stationary random measures. These induce a natural notion of interpolation…
Gaussian random fields on finite dimensional smooth manifolds whose variances reach their maximum value at smooth submanifolds are considered. Exact asymptotic behaviors of large excursion probabilities have been evaluated. Vector Gaussian…
The independent interval approximation of the excursion time distributions for Gaussian processes has been used in physics and engineering. A new but related approach matches the expected value of the clipped Slepian to the expected value…
We present a new method to compute the first crossing distribution in excursion set theory for the case of correlated random walks. We use a combination of the path integral formalism of Maggiore & Riotto, and the integral equation solution…
Let $X = \{X(t): t\in T \}$ be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space $T$, and let $A_u(X,T) = \{t\in T: X(t)\geq u\}$ be the excursion set of $X$ exceeding level $u$. Under certain…
This paper studies Gaussian random fields with Mat\'ern covariance functions with smooth parameter $\nu>2$. Two cases of parameter spaces, the Euclidean space and $N$-dimensional sphere, are considered. For such smooth Gaussian fields, we…
We develop a novel computational method for evaluating the extreme excursion probabilities arising from random initialization of nonlinear dynamical systems. The method uses excursion probability theory to formulate a sequence of Bayesian…
The goal of this paper is to give confidence regions for the excursion set of a spatial function above a given threshold from repeated noisy observations on a fine grid of fixed locations. Given an asymptotically Gaussian estimator of the…
We propose a method of estimating the uncertainty of a result obtained through extrapolation to the complete basis set limit. The method is based on an ensemble of random walks which simulate all possible extrapolation outcomes that could…
We describe the first known mean-field study of landing probabilities for random walks on hypergraphs. In particular, we examine clique-expansion and tensor methods and evaluate their mean-field characteristics over a class of random…
Relying on the excursion set theory, we compute the number density of local extrema and crossing statistics versus the threshold for the stock market indices. Comparing the number density of excursion sets calculated numerically with the…
We consider the problem of model selection in Gaussian Markov fields in the sample deficient scenario. In many practically important cases, the underlying networks are embedded into Euclidean spaces. Using the natural geometric structure,…
We provide a simple formula that accurately approximates the first crossing distribution of barriers having a wide variety of shapes, by random walks with a wide range of correlations between steps. Special cases of it are useful for…
Let $\{X_i(t):\, t\in S\subset \R^d \}_{i=1,2,\ldots,n}$ be independent copies of a stationary centered Gaussian field with almost surely smooth sample paths. In this paper, we are interested in the conjunction probability defined as $\PP…
Random field excursions is an increasingly vital topic within data analysis in medicine, cosmology, materials science, etc. This work is the first detailed study of their Betti numbers in the so-called `sparse' regime. Specifically, we…
We have obtained some upper bounds for the probability distribution of extremes of a self-similar Gaussian random field with stationary rectangular increments that are defined on the compact spaces. The probability distributions of extremes…
A flexible model for non-stationary Gaussian random fields on hypersurfaces is introduced.The class of random fields on curves and surfaces is characterized by an amplitude spectral density of a second order elliptic differential…