Related papers: Optimal quantization for nonuniform Cantor distrib…
In the first part of this study (Paper I), we introduced the systematic improvement probability (SIP) as a tool to assess the level of improvement on absolute errors to be expected when switching between two computational chemistry methods.…
In this article, we study the approximation of a probability measure $\mu$ on $\mathbb{R}^{d}$ by its empirical measure $\hat{\mu}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment…
The problem of quantization of measures looks for best approximations of probability measures on a metric space by discrete measures supported on $N$ points, where the error of approximation is measured with respect to the Wasserstein…
A common way to discretize a probability measure is to use an empirical measure as a discrete approximation. But how far from being optimal is this approximation in the p-Wasserstein distance? In this paper, we study this question in two…
Consider a photon that has just emerged from a linear polarizing filter. If the photon is then subjected to an orthogonal polarization measurement-e.g., horizontal vs vertical-the photon's preparation cannot be fully expressed in the…
We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is…
Predicting the outcomes of quantum measurements is a cornerstone of quantum information theory and a key resource for quantum technologies. Here, we introduce a comprehensive framework for quantifying the predictability of measurements on a…
This paper presents a detailed study of constrained quantization for both finite and infinite discrete probability distributions supported on subsets of the real line. Under specific geometric constraints - namely, a semicircular arc and…
We consider statistical learning problems in which data are observed as a set of probability measures. Optimal transport (OT) is a popular tool to compare and manipulate such objects, but its computational cost becomes prohibitive when the…
We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.
It is known that for a $\rho$-weighted $L_q$-approximation of single variable functions $f$ with the $r$th derivatives in a $\psi$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to…
We provide a complete picture of the upper quantization dimension in terms of the R\'enyi dimension by proving that the upper quantization dimension $\bar{D}_{r}(\nu)$ of order $r>0$ for an arbitrary compactly supported Borel probability…
Based on the seminal work of Hutchinson, we investigate properties of {\em $\alpha$-weighted Cantor measures} whose support is a fractal contained in the unit interval. Here, $\alpha$ is a vector of nonnegative weights summing to $1$, and…
For any Borel probability measure on $\mathbb{R}^n$, we may define a family of eccentricity tensors. This new notion, together with a tensorization trick, allows us to prove an energy minimization property for rotationally invariant…
In this paper, we are going to discuss the following problem: Let $T$ be a fixed set in $\mathbb{R}^n$. And let $S$ and $B$ he two subsets in $\mathbb{R}^n$ such that for any $x$ in $S$, there exists an $r$ such that $x+ r T$ is a subset of…
We introduce a 'uniform tension-reduction' (UTR) model, which allows to represent the probabilities associated with an arbitrary measurement situation and use it to explain the emergence of quantum probabilities (the Born rule) as 'uniform'…
This paper is devoted to the stochastic approximation of entropically regularized Wasserstein distances between two probability measures, also known as Sinkhorn divergences. The semi-dual formulation of such regularized optimal…
We study the problem of testing discrete distributions with a focus on the high probability regime. Specifically, given samples from one or more discrete distributions, a property $\mathcal{P}$, and parameters $0< \epsilon, \delta <1$, we…
With the forthcoming release of high precision polarization measurements, such as from the Planck satellite, it becomes critical to evaluate the performance of estimators for the polarization fraction and angle. These two physical…
Counting experiments often rely on Monte Carlo simulations for predictions of Poisson expectations. The accompanying uncertainty from the finite Monte Carlo sample size can be incorporated into parameter estimation by modifying the Poisson…