Related papers: Quantization for uniform distributions on equilate…
We establish general conditions under which there exists uniform in time convergence between a stochastic process and its approximated system. These standardised conditions consist of a local in time estimate between the original and the…
Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). We show that if there exists a sequence of positive real numbers $(t_n)_{n=1}^{\infty}$ converging to 0 such that $$\lim_{N \rightarrow \infty}{…
We analyze the problem of sending, in a single transmission, the information required to specify an orthogonal trihedron or reference frame through a quantum channel made out of N elementary spins. We analytically obtain the optimal…
We give a characterization of all three points in $\mathbb R^4$ with integer coordinates which are at the same Euclidean distance apart. In three dimension the problem is characterized in terms of solutions of the Diophantine equations…
We prove that uniform random triangulations whose genus is proportional to their size $n$ have diameter of order $\log n$ with high probability. We also show that in such triangulations, the distances between most pairs of points differ by…
Uniform distribution of the points has been of interest to researchers for a long time and has applications in different areas of Mathematics and Computer Science. One of the well-known measures to evaluate the uniformity of a given…
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…
We study the problem of distributed mean estimation and optimization under communication constraints. We propose a correlated quantization protocol whose leading term in the error guarantee depends on the mean deviation of data points…
A reliable method for characterizing quantum operations that is suitable for improving and validating their accuracies is indispensable for realizing a practical quantum computer. Known methods are still not sufficient because they lack…
We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.
Finite precision approximations of discrete probability distributions are considered, applicable for distribution synthesis, e.g., probabilistic shaping. Two algorithms are presented that find the optimal $M$-type approximation $Q$ of a…
The explicit incorporation of task-specific inductive biases through symmetry has emerged as a general design precept in the development of high-performance machine learning models. For example, group equivariant neural networks have…
Sampling and quantization are crucial in digital signal processing, but quantization introduces errors, particularly due to distribution mismatch between input signals and quantizers. Existing methods to reduce this error require precise…
This article studies a general divide-and-conquer algorithm for approximating continuous one-dimensional probability distributions with finite mean. The article presents a numerical study that compares pre-existing approximation schemes…
Almost $50$ years ago Erd\H{o}s and Purdy asked the following question: Given $n$ points in the plane, how many triangles can be approximate congruent to equilateral triangles? They pointed out that by dividing the points evenly into three…
We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x…
Science about optimization methods is rapidly developing today. In machine learning, computer vision, biology, medicine, construction and in many other different areas optimization methods have vast popularity and they appear as important…
We consider a diffusion equation with highly oscillatory coefficients that admits a homogenized limit. As an alternative to standard corrector problems, we introduce here an embedded corrector problem, written as a diffusion equation in the…
We address the experimental determination of entanglement for systems made of a pair of polarization qubits. We exploit quantum estimation theory to derive optimal estimators, which are then implemented to achieve ultimate bound to…
Error correcting codes with a universal set of transversal gates are a desideratum for quantum computing. Such codes, however, are ruled out by the Eastin-Knill theorem. Moreover, the theorem also rules out codes which are covariant with…