Related papers: Hadwiger Integration of Random Fields
Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable…
We give a characterization of smooth, rotation and dually epi-translation invariant valuations and use this result to obtain a new proof of the Hadwiger theorem on convex functions. We also give a description of the construction of the…
A new version of the Hadwiger theorem on convex functions is established and an explicit representation of functional intrinsic volumes is found using new functional Cauchy-Kubota formulas. In addition, connections between functional…
A complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on ${\mathbb R}^n$ is established. The valuations obtained are functional versions of the…
In this paper, we derive tail approximations of integrals of exponential functions of Gaussian random fields with varying mean functions and approximations of the associated point processes. This study is motivated naturally by multiple…
We explore a well-known integral representation of the logarithmic function, and demonstrate its usefulness in obtaining compact, easily-computable exact formulas for quantities that involve expectations and higher moments of the logarithm…
Intrinsic volumes, which generalize both Euler characteristic and Lebesgue volume, are important properties of $d$-dimensional sets. A random cubical complex is a union of unit cubes, each with vertices on a regular cubic lattice,…
In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to…
Various applications ranging from robotics to climate science require modeling signals on non-Euclidean domains, such as the sphere. Gaussian process models on manifolds have recently been proposed for such tasks, in particular when…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge-Amp\`ere measures and a new version of the Hadwiger theorem on convex…
A complete classification of continuous, dually epi-translation invariant, and rotation equivariant valuations on convex functions is established. This characterizes the recently introduced functional Minkowski vectors, which naturally…
The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random…
The famous Hadwiger theorem classifies all rigid motion invariant continuous valuations on convex sets as linear conbinations of quermassintegrals. We prove much more general result. We classify continuous valuations which are invariant…
Gaussian processes (GPs) defined through intrinsic random fields provide a flexible framework for modeling spatial phenomena, and have been advocated in a variety of applications over the past several decades. Nevertheless, their adoption…
We develop a framework for generalized variational inference in infinite-dimensional function spaces and use it to construct a method termed Gaussian Wasserstein inference (GWI). GWI leverages the Wasserstein distance between Gaussian…
We define the notion of valuation on simplicial maps between geometric realizations of simplicial complexes in $\mathbb{R}^n$. Valuations on simplicial maps are analogous to valuations on sets. In particular, we define the Lefschetz…
Context: Two-point correlation functions are used throughout cosmology as a measure for the statistics of random fields. When used in Bayesian parameter estimation, their likelihood function is usually replaced by a Gaussian approximation.…
An algorithm for integration of polynomial functions with variable weight is considered. It provides extension of the Gaussian integration, with appropriate scaling of the abscissas and weights. Method is a good alternative to usually…
In this paper, we discuss vector-valued Gaussian processes for the approximation of divergence- or rotation-free functions. We establish the theory for such Gaussian processes, then link the theory to multivariate approximation theory, and…