Related papers: Information-Theoretic Inequalities on Unimodular L…

Information geometry provides differential geometric concepts like a Riemannian metric, connections and covariant derivatives on spaces of probability distributions. We discuss here how these concepts apply to quantum field theories in the…

Classical mechanics unfolds within absolute time and Euclidean space, yet our knowledge of where events occur, when they occur, and how motion evolves is inherently uncertain. The special Galilean group provides a natural setting for…

In this work, we present straightforward and concrete computations of the unitary irreducible representations of the Euclidean motion group $M(2)$ employing the methods of deformation quantization. Deformation quantization is a quantization…

Change of measure inequalities translate divergences between probability measures into explicit bounds on event probabilities, and play an important role in deriving probabilistic guarantees in learning theory, information theory, and…

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. We view these results as uncertainty…

Symmetry principles underlie and guide scientific theory and research, from Curie's invariance formulation to modern applications across physics, chemistry, and mathematics. Building on a recent matrix Lie group measurement model, this…

Crystallographic groups describe the symmetries of crystals and other repetitive structures encountered in nature and the sciences. These groups include the wallpaper and space groups. We derive linear and nonlinear representations of…

Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie…

In a series of works, Lutwak, Yang and Zhang established what could be called affine information theory, which is the study of moment-entropy and Fisher-information-type inequalities that are invariant with respect to affine transformations…

We present a unified approach to obtain Hardy-type inequalities in the context of nilpotent Lie groups with sharp constants. The unified methodology employed herein allows for exploration of the sharp Hardy inequalities on various Lie group…

While most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information, up to now Shannon's entropy power inequality (EPI) is an exception: Existing information theoretic proofs of the…

This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that…

In statistical inference, uncertainty is unknown and all models are wrong. That is to say, a person who makes a statistical model and a prior distribution is simultaneously aware that both are fictional candidates. To study such cases,…

Data sets sampled in Lie groups are widespread, and as with multivariate data, it is important for many applications to assess the differences between the sets in terms of their distributions. Indices for this task are usually derived by…

This paper addresses uncertainty propagation on unimodular matrix Lie groups that have a surjective exponential map. We derive the exact formula for the propagation of mean and covariance in a continuous-time setting from the governing…

This paper proposes a unifying variational approach for proving and extending some fundamental information theoretic inequalities. Fundamental information theory results such as maximization of differential entropy, minimization of Fisher…

We consider the problems of clustering, classification, and visualization of high-dimensional data when no straightforward Euclidean representation exists. Typically, these tasks are performed by first reducing the high-dimensional data to…

We carry out a systematic study of uncertainty measures that are generic to dynamical processes of varied origins, provided they induce suitable continuous probability distributions. The major technical tool are the information theory…

The transition to Euclidean space and the discretization of quantum field theories on spatial or space-time lattices opens up the opportunity to investigate probabilistic machine learning within quantum field theory. Here, we will discuss…

A Lie group is an old mathematical abstract object dating back to the XIX century, when mathematician Sophus Lie laid the foundations of the theory of continuous transformation groups. As it often happens, its usage has spread over diverse…