Related papers: Theory of approximation and continuity of random p…

Deep neural networks (DNN) and Gaussian processes (GP) are two powerful models with several theoretical connections relating them, but the relationship between their training methods is not well understood. In this paper, we show that…

One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in…

These are classified by the direction of approximation (from above or below), the set family types (partition or covering) of simple functions, the coefficient signature (non-negative or signed), and cardinal number of terms of simple…

Recently, Freud-Rodriguez proposed a new counting process which is called the Bell-Touchard process and based on the Bell-Touchard probability distribution. This process was developed to solve the problem of rare events hypothesis which is…

A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory…

Rough set theory is a new mathematical approach to imperfect knowledge. The notion of rough sets is generalized by using an arbitrary binary relation on attribute values in information systems, instead of the trivial equality relation. The…

Scaling analysis, in which one infers scaling exponents and a scaling function in a scaling law from given data, is a powerful tool for determining universal properties of critical phenomena in many fields of science. However, there are…

We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the…

The non-local degenerate Cahn-Hilliard equation is derived from the Vlasov equation with long-range attraction. We study the local limit as the delocalization parameter converges to 0. The difficulty arises from the degeneracy which…

One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and non-stationary regimes is the lack of a suitable limit object. Instead, suitable distributional approximations can be used, where…

We introduce a framework to derive quantitative central limit theorems in the context of non-linear approximation of Gaussian random variables taking values in a separable Hilbert space. In particular, our method provides an alternative to…

We study moderate deviations of suprema of parametrized sequences of sample bounded Gaussian processes $\{X _x(t), t\in T _x\}$, and first present recent sharp bounds in simple cases. In the almost periodic case, we prove an approximation…

Variational inference is a powerful tool for approximate inference, and it has been recently applied for representation learning with deep generative models. We develop the variational Gaussian process (VGP), a Bayesian nonparametric…

We develop a pseudo-metric analogue of bisimulation for generalized semi-Markov processes. The kernel of this pseudo-metric corresponds to bisimulation; thus we have extended bisimulation for continuous-time probabilistic processes to a…

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as…

We derive the necessary and sufficient condition for almost sure convergence of the sequence of measurable functions, and consider some applications in the theory of Fourier series and in the theory of random fields.

We develop some theoretical results for a robust similarity measure named "generalized min-max" (GMM). This similarity has direct applications in machine learning as a positive definite kernel and can be efficiently computed via…

This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling -- a central task in statistical…

Random events in space and time often exhibit a locally dependent structure. When the events are very rare and dependent structure is not too complicated, various studies in the literature have shown that Poisson and compound Poisson…

Random feature approximation is arguably one of the most popular techniques to speed up kernel methods in large scale algorithms and provides a theoretical approach to the analysis of deep neural networks. We analyze generalization…