Related papers: The Marcinkiewicz-type discretization theorems
Topological measurements are increasingly being accepted as an important tool for quantifying complex structures. In many applications, these structures can be expressed as nodal domains of real-valued functions and are obtained only…
The discretization of the density matrix is proposed as a nonlinear positive map for systems with continuous variables. This procedure is used to calculate the entanglement between two modes through different criteria, such as Tsallis…
Some general considerations on the notion of entropy in physics are presented. An attempt is made to clarify the question of the differentiation between physical entropy (the Clausius-Boltzmann one) and quantities called entropies…
Tools from optimal transport (OT) theory have recently been used to define a notion of quantile function for directional data. In practice, regularization is mandatory for applications that require out-of-sample estimates. To this end, we…
Reliable data-driven estimation of Shannon entropy from small data sets, where the number of examples is potentially smaller than the number of possible outcomes, is a critical matter in several applications. In this paper, we introduce a…
We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming…
We study in this paper a multilayer discretization of second order elliptic problems, aimed at providing reliable multilayer discretizations of shallow fluid flow problems with diffusive effects. This discretization is based upon the…
In the Normed space theory, the existence of fixed points is one of the main tools in improving efficiency of iterative algorithms in optimization, numerical analysis and various mathematical applications. This study introduces and…
Statistical mechanics is generalized on the basis of an information theory for inexact or incomplete probability distributions. A parameterized normalization is proposed and leads to a nonextensive entropy. The resulting incomplete…
Uncertainty quantification is a key aspect in many tasks such as model selection/regularization, or quantifying prediction uncertainties to perform active learning or OOD detection. Within credal approaches that consider modeling…
The profile of a sample is the multiset of its symbol frequencies. We show that for samples of discrete distributions, profile entropy is a fundamental measure unifying the concepts of estimation, inference, and compression. Specifically,…
We introduce a framework for uncertainty estimation that both describes and extends many existing methods. We consider typical hyperparameters involved in classical training as random variables and marginalise them out to capture various…
We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of $d$-dimensional bounded monotonic functions under $L^p$ norms. It is interesting to see that both the metric entropy and bracketing entropy…
We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope…
In this work, a generalization of the well known Bernoulli inequality is obtained by using the theory of discrete fractional calculus. As far as we know our approach is novel.
One of the fundamental problems in machine learning is the estimation of a probability distribution from data. Many techniques have been proposed to study the structure of data, most often building around the assumption that observations…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…
We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in…
A novel, non-trivial, probabilistic upper bound on the entropy of an unknown one-dimensional distribution, given the support of the distribution and a sample from that distribution, is presented. No knowledge beyond the support of the…
We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means,…