Related papers: Krivine schemes are optimal

By a classical result of Gauss and Kuzmin, the frequency with which a string $\mathbf{a}=(a_1,\dots,a_n)$ of positive integers appears in the continued fraction expansion of a random real number is given by $\mu_{GK}({I(\mathbf{a})})$,…

The maximum likelihood method is the best-known method for estimating the probabilities behind the data. However, the conventional method obtains the probability model closest to the empirical distribution, resulting in overfitting. Then…

We establish a near-optimality guarantee for the full orthogonalization method (FOM), showing that the overall convergence of FOM is nearly as good as GMRES. In particular, we prove that at every iteration $k$, there exists an iteration…

It is shown that the homogeneous ergodic bilinear averages with M\"{o}bius or Liouville weight converge almost surely to zero, that is, if $T$ is a map acting on a probability space $(X,\mathcal{A},\mu)$, and $a,b \in \mathbb{Z}$, then for…

This paper investigates regularized estimation of Kronecker-structured covariance matrices (CM) for polarization radar in sea clutter scenarios where the data are assumed to follow the complex, elliptically symmetric (CES) distributions…

Comparing probability distributions is an indispensable and ubiquitous task in machine learning and statistics. The most common way to compare a pair of Borel probability measures is to compute a metric between them, and by far the most…

We prove that for every $c\in(1,23/22)$, every probability space $(X,\mathcal{B},\mu)$ equipped with two commuting measure-preserving transformations $T,S\colon X\to X$ and every $f,g\in L^{\infty}_{\mu}(X)$ we have that the…

A $k$-tuple $(H_1, \ldots, H_k)$ of core-free subgroups of a finite group $G$ is said to be regular if $G$ has a regular orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_k$. The regularity number of $G$, denoted $R(G)$, is the…

We develop some theoretical results for a robust similarity measure named "generalized min-max" (GMM). This similarity has direct applications in machine learning as a positive definite kernel and can be efficiently computed via…

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1 \le p \neq n$. We prove that, for every pair of probability measures $\mu,\nu$ satisfying a…

Let g(x)=x/2 + 17/30 (mod 1), let \xi_i, i= 1,2,... be a sequence of independent, identically distributed random variables with uniform distribution on the interval [0,1/15], define g_i(x)=g(x)+ \xi_i (mod 1) and, for n=1,2,..., define…

In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix appearing in the circulant embedding method is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail…

In a previous article, given a finite-dimensional real vector space $V$ and a probability measure $\mu$ on $\operatorname{PGL}(V)$ with finite first moment, we gave a description of all $\mu$-stationary probability measures on the…

Let $M_n$ denote a random symmetric $n \times n$ matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely…

Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow…

Let $I, J\subset \mathbb{R}$ be closed intervals, and let $H$ be $C^{3}$ smooth real valued function on $I\times J$ with nonvanishing $H_{x}$ and $H_{y}$. Take any fixed positive numbers $a,b$, and let $d\mu$ be a probability measure with…

Let $\nu$ be a Borel probability measure on a $d$-dimensional Euclidean space $\mathbb{R}^d$, $d\geq 1$, with a compact support, and let $(p_0, p_1, p_2, \ldots, p_N)$ be a probability vector with $p_j>0$ for $0\leq j\leq N$. Let $\{S_j:…

We consider an $N$ by $N$ real symmetric random matrix $X=(x_{ij})$ where $\mathbb{E}x_{ij}x_{kl}=\xi_{ijkl}$. Under the assumption that $(\xi_{ijkl})$ is the discretization of a piecewise Lipschitz function and that the correlation is…

This paper concerns the approximation of probability measures on $\mathbf{R}^d$ with respect to the Kullback-Leibler divergence. Given an admissible target measure, we show the existence of the best approximation, with respect to this…

Given a possibly discontinuous, bounded function $f:\mathbb{R}\mapsto\mathbb{R}$, we consider the set of generalized flows, obtained by assigning a probability measure on the set of Carath\'eodory solutions to the ODE ~$\dot x = f(x)$. The…