Related papers: On the symbol error probability of regular polytop…

This paper considers the problem of recovering the permutation of an n-dimensional random vector X observed in Gaussian noise. First, a general expression for the probability of error is derived when a linear decoder (i.e., linear estimator…

Future communication systems are anticipated to facilitate applications requiring high data transmission rates while maintaining energy efficiency. Hexagonal quadrature amplitude modulation (HQAM) offers this owing to its compact symbol…

The symbol error rate of the minimum distance detector for an arbitrary multi-dimensional constellation impaired by additive white Gaussian noise is characterized as the product of a completely monotone function with a non-negative power of…

Let $X_1,\ldots,X_n$ be a standard normal sample in $\mathbb R^d$. We compute exactly the expected volume of the Gaussian polytope $\mathrm{conv}[X_1,\ldots,X_n]$, the symmetric Gaussian polytope $\mathrm{conv}[\pm X_1,\ldots,\pm X_n]$, and…

We derive explicit formulae for the expected volume and the expected number of facets of the convex hull of several multidimensional Gaussian random walks in terms of the Gaussian persistence probabilities. Special cases include the already…

The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of $n$ independent standard normally distributed points in $\mathbb R^d$. We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point…

We consider geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise. One famous questions in the field is the "weak simplex conjecture". We discuss possible approaches to it, and…

Consider the random polytope, that is given by the convex hull of a Poisson point process on a smooth convex body in $\mathbb{R}^d$. We prove central limit theorems for continuous motion invariant valuations including the Will's functional…

To complete a previous work, the probability density functions for the errors in the center-of-gravity as positioning algorithm are derived with the usual methods of the cumulative distribution functions. These methods introduce substantial…

A popular approach for estimating an unknown signal from noisy, linear measurements is via solving a so called \emph{regularized M-estimator}, which minimizes a weighted combination of a convex loss function and of a convex (typically,…

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic…

We make an imaginative comparison between the Minimal Supersymmetric Standard Model and the 24-cell polytope in four dimensions, the Octacube.

We derive a heuristic nonlinear interference model for 4D probabilistic shaping considering the polarization and time correlation of the 4D symbols. We demonstrate an average SNR prediction gap from split-step Fourier simulations of…

In this paper we review different expansions for neutrino oscillation probabilities in matter in the context of long-baseline neutrino experiments. We examine the accuracy and computational efficiency of different exact and approximate…

This paper provides full \Matlab-code and informal correctness proofs for the lexicographic reverse search algorithm for convex hull calculations. The implementation was tested on a 1993 486-PC for various small and some larger, partially…

This paper develops asymptotic methods to count faces of random high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have surprising implications - in statistics, probability, information theory, and signal processing…

We study the regularity of densities of distributions that are polynomial images of the standard Gaussian measure on $\mathbb{R}^n$. We assume that the degree of a polynomial is fixed and that each variable enters to a power bounded by…

The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation…

The convex hull $P_{n}$ of a Gaussian sample $X_{1},...,X_{n}$ in $R^{d}$ is a Gaussian polytope. We prove that the expected number of facets $E f_{d-1} (P_n)$ is monotonically increasing in $n$. Furthermore we prove this for random…

The intrinsic volumes of Gaussian polytopes are considered. A lower variance bound for these quantities is proved, showing that, under suitable normalization, the variances converge to strictly positive limits. The implications of this…