Related papers: Information Theory and Quadrature Rules

As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the…

The information spectrum approach gives general formulae for optimal rates of codes in many areas of information theory. In this paper the quantum spectral divergence rates are defined and properties of the rates are derived. The entropic…

In this paper we study the numerical quadrature of a stochastic integral, where the temporal regularity of the integrand is measured in the fractional Sobolev-Slobodeckij norm in $W^{\sigma,p}(0,T)$, $\sigma \in (0,2)$, $p \in [2,\infty)$.…

We consider quadrature formulas based on interpolation using the basis functions $1/(1+t_kx)$ $(k=1,2,3,\ldots)$ on $[-1,1]$, where $t_k$ are parameters on the interval $(-1,1)$. We investigate two types of quadratures: quadrature formulas…

We consider the computation of quadrature rules that are exact for a Chebyshev set of linearly independent functions on an interval $[a,b]$. A general theory of Chebyshev sets guarantees the existence of rules with a Gaussian property, in…

The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for finite discrete…

In this paper, we consider a zero-order stochastic oracle model of estimating definite integrals. In this model, integral estimation methods may query an oracle function for a fixed number of noisy values of the integrand function and use…

The uncertainty principle can be expressed in entropic terms, also taking into account the role of entanglement in reducing uncertainty. The information exclusion principle bounds instead the correlations that can exist between the outcomes…

We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a…

Highly oscillatory integrals of composite type arise in electronic engineering and their calculations is a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is…

It is shown that quadrature formulas in many different applications can be derived from rational approximation of the Cauchy transform of a weight function. Since rational approximation is now a routine technology, this provides an easy new…

In order to approximate the integral $I(f)=\int_a^b f(x) dx$, where $f$ is a sufficiently smooth function, models for quadrature rules are developed using a given {\it panel} of $n (n\geq 2)$ equally spaced points. These models arise from…

In this paper, a unified framework for representing uncertain information based on the notion of an interval structure is proposed. It is shown that the lower and upper approximations of the rough-set model, the lower and upper bounds of…

Information theory provides tools to predict the performance of a learning algorithm on a given dataset. For instance, the accuracy of learning an unknown parameter can be upper bounded by reducing the learning task to hypothesis testing…

Driven by several successful applications such as in stochastic gradient descent or in Bayesian computation, control variates have become a major tool for Monte Carlo integration. However, standard methods do not allow the distribution of…

There are (at least) three approaches to quantifying information. The first, algorithmic information or Kolmogorov complexity, takes events as strings and, given a universal Turing machine, quantifies the information content of a string as…

Quantum error-correcting codes are analyzed from an information-theoretic perspective centered on quantum conditional and mutual entropies. This approach parallels the description of classical error correction in Shannon theory, while…

In the classic scoring rule setting, a principal incentivizes an agent to truthfully report their probabilistic belief about some future outcome. This paper addresses the situation when this private belief, rather than a classical…

For the class of polynomial quadrature rules we show that conveniently chosen bases allow to compute both the weights and the theoretical error expression of a $n$-point rule via the undetermined coefficients method. As an illustration, the…

We generalize proper scoring rules to the quantum domain, replacing probability distributions with density operators. We define Quantum Value Functionals via operator convex generators and establish a complete duality theory yielding proper…