Related papers: Estimation of arithmetic linear series

Our goal is to finally settle the persistent problem in Diophantine Approximation of finding best linear approximates. Classical results from the theory of continued fractions provide the solution for the special homogeneous case in the…

In this work we consider sums of primes that converging very slow. We set as a base, a reformulation of analytic prime number theorem and we use the values of Riemann Zeta function for the approximation. We also give the truncation error of…

We establish a quantitative approximation formula of the Lyapunov exponent of a rational function of degree more than one over an algebraically closed field of characteristic $0$ that is complete with respect to a non-trivial and possibly…

Motivated by the groundbreaking work of Andrews and Merca, truncated theta series have been extensively studied over the years. In particular, Merca made conjectures on the non-negativity of the coefficient of $q^N$ in truncated series from…

Matrix completion algorithms recover a low rank matrix from a small fraction of the entries, each entry contaminated with additive errors. In practice, the singular vectors and singular values of the low rank matrix play a pivotal role for…

We deduce Diophantine arithmetic inequalities for big linear systems and with respect to finite extensions of number fields. Our starting point is the Parametric Subspace Theorem, for linear forms, as formulated by Evertse and Ferretti…

The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of…

In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan, who proved a Central Limit Theorem for the usual least squares estimator under…

Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of…

We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprind\v{z}uk from…

The paper considers a universal approach that allows one to quite simply obtain nonlinear asymptotic estimates of various summation functions. It is shown the application of this approach to the asymptotic estimation of divergent Dirichlet…

We refine the classical Lindeberg-Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parametrized Prokhorov distances in terms of a Lindeberg index. We thus obtain more…

We study adaptive approximation algorithms for general multivariate linear problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform essentially no better than non-adaptive…

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…

We derive a series summation formula for the average logarithm norm of the action of a matrix on the projective space. This formula is shown to be useful to evaluate some Lyapunov exponents of random $\SL$-matrix cocycles, which include a…

We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006b) allows to express the exact partition function Z of a graphical model as a…

We consider the marginal models of Liang and Zeger [Biometrika 73 (1986) 13-22] for the analysis of longitudinal data and we develop a theory of statistical inference for such models. We prove the existence, weak consistency and asymptotic…

We study the estimation of the linear discriminant with projection pursuit, a method that is blind in the sense that it does not use the class labels in the estimation. Our viewpoint is asymptotic and, as our main contribution, we derive…

The majority of machine learning methods can be regarded as the minimization of an unavailable risk function. To optimize the latter, given samples provided in a streaming fashion, we define a general stochastic Newton algorithm and its…

The chief aim of this paper is to propose mean-field approximations for a broad class of Belief networks, of which sigmoid and noisy-or networks can be seen as special cases. The approximations are based on a powerful mean-field theory…