Related papers: Hyperuniformity and hyperfluctuations of random me…
Consider a positive Borel measure on a locally compact group. We define a notion of uniform density for such a measure, which is based on a group invariant introduced by Leptin in 1966. We then restrict to unimodular amenable groups and to…
Let X be a real or complex Hilbert space of finite but large dimension d, let S(X) denote the unit sphere of X, and let u denote the normalized uniform measure on S(X). For a finite subset B of S(X), we may test whether it is approximately…
On a compact group the Haar probability measure plays the role of uniform distribution. The entropy and rate distortion theory for this uniform distribution is studied. New results and simplified proofs on convergence of convolutions on…
Let $\{U^N_t\}_{t\ge 0}$ be a standard Brownian motion on $\mathbb{U}(N)$. For fixed $N\in\mathbb{N}$ and $t>0$, we give explicit bounds on the $L_1$-Wasserstein distance of the empirical spectral measure of $U^N_t$ to both the…
The measurement processes that are traditionally described within the realm of non-relativistic quantum mechanics are transcribed into the covariant framework of Cartan's space, the four-valued representation space of the restricted…
Data represented by probability measures arise as empirical distributions, posterior distributions, and feature-based representations of complex objects. We study heterogeneity in a population of probability measures through the expected…
By establishing the intrinsic super-Poincar\'e inequality, some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the…
We propose a new probabilistic characterization of the uniform distribution on the hypersphere in terms of the distribution of pairwise inner products, extending the ideas of \citep{cuesta2009projection,cuesta2007sharp} in a data-driven…
We show that for ultracontractive irreducible Dirichlet metric measure spaces, the Dirichlet spectrum is discrete for a restriction to any connected open set without any assumption on regularity of the boundary. The main applications…
We describe the spectrum of an ergodic invariant measure by examining the behaviour of its generic points. We define regular Wiener--Wintner generic points for a measure to generalise the characterisation of generic points for discrete…
We introduce and investigate in this short report the new notion of uniform measure (distribution) on the arbitrary compact metric space. We consider also some possible applications of these measures in the theory of imbedding theorems and…
Dyadic data is often encountered when quantities of interest are associated with the edges of a network. As such it plays an important role in statistics, econometrics and many other data science disciplines. We consider the problem of…
It is shown to be consistent with set theory that the uniformity invariant for Lebesgue measure is strictly greater than the corresponding invariant for Hausdorff r-dimensional measure where 0<r<1.
We generalize the recent results of Bj\"orn-Bj\"orn-Shanmugalingam \cite{BBS20} concerning how measures transform under the uniformization procedure of Bonk-Heinonen-Koskela for Gromov hyperbolic spaces \cite{BHK} by showing that these…
To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of $\mathbb{R}^d$. For restrictions to the Euclidean ball in odd…
We consider vectors of random variables, obtained by restricting the length of the nodal set of Berry's random wave model to a finite collection of (possibly overlapping) smooth compact subsets of $\mathbb{R}^2$. Our main result shows that,…
Assuming the heat kernel on a doubling Dirichlet metric measure space has a sub-Gaussian bound, we prove an asymptotically sharp spectral upper bound on the survival probability of the associated diffusion process. As a consequence, we can…
An analysis of quantum measurement is presented that relies on an information-theoretic description of quantum entanglement. In a consistent quantum information theory of entanglement, entropies (uncertainties) conditional on measurement…
In many contemporary statistical and machine learning methods, one needs to optimize an objective function that depends on the discrepancy between two probability distributions. The discrepancy can be referred to as a metric for…
Boolean formulae compactly encode huge, constrained search spaces. Thus, variability-intensive systems are often encoded with Boolean formulae. The search space of a variability-intensive system is usually too large to explore without…