arXiv subjects

Miklós Kornyik

Publications and source records attributed to Miklós Kornyik.

4 recordsLinked to original sources

arXiv2025

Low complexity convergence rate bounds for push-sum algorithms with homogeneous correlation structure

The objective of this work is to establish an upper bound for the almost sure convergence rate for a class of push-sum algorithms. The current work extends the methods and results of the authors on a similar low-complexity bound on push-sum algorithms with some particular synchronous message passing schemes and complements the general approach of Gerencsér and Gerencsér from 2022 providing an exact, but often less accessible description. Furthermore, a parametric analysis is presented on the ``weight'' of the messages, which is found to be convex with an explicit expression for the gradient. This allows the fine-tuning of the algorithm used for improved efficiency. Numerical results confirm the speedup in evaluating the computable bounds without deteriorating their performance, for a graph on 120 vertices the runtime drops by more than 4 orders of magnitude.

math.PR↗

arXiv2023

Low complexity convergence rate bounds for the synchronous gossip subclass of push-sum algorithms

We develop easily accessible quantities for bounding the almost sure exponential convergence rate of push-sum algorithms. We analyze the scenario of i.i.d. synchronous gossip, every agent communicating towards its single target at every step. Multiple bounding expressions are developed depending on the generality of the setup, all functions of the spectrum of the network. While the most general bound awaits further improvement, with more symmetries, close bounds can be established, as demonstrated by numerical simulations.

math.PR↗

arXiv2016

Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law

In the present paper we give two alternate proofs of the well known theorem that the empirical distribution of the appropriately normalized roots of the $n^{th}$ monic Hermite polynomial $H_n$ converges weakly to the semicircle law, which is also the weak limit of the empirical distribution of appropriately normalized eigenvalues of a Wigner matrix. In the first proof -- based on the recursion satisfied by the Hermite polynomials -- we show that the generating function of the moments of roots of $H_n$ is convergent and it satisfies a fixed point equation, which is also satisfied by $c(z^2)$, where $c(z)$ is the generating function of the Catalan numbers $C_k$. In the second proof we compute the leading and the second leading term of the $k^{th}$ moments (as a polynomial in $n$) of $H_n$ and show that the first one coincides with $C_{k/2}$, the $(k/2)^{\rm th}$ Catalan number, where $k$ is even and the second one is given by $-(2^{2k-1}-\binom{2k-1}{k})$. We also mention the known result that the expectation of the characteristic polynomial ($p_n$) of a Wigner random matrix is exactly the Hermite polynomial ($H_n$), i.e. $Ep_n(x)=H_n(x)$, which suggest the presence of a deep connection between the Hermite polynomials and Wigner matrices.

math.CA↗

arXiv2016

On the moments of roots of Laguerre-polynomials and the Marchenko-Pastur law

In this paper we compute the leading terms in the sum of the $k^{th}$ power of the roots of $L_{p}^{(α)}$, the Laguerre-polynomial of degree $p$ with parameter $α$. The connection between the Laguerre-polynomials and the Marchenko-Pastur distribution is expressed by the fact, among others, that the limiting distribution of the empirical distribution of the normalized roots of the Laguerre-polynomials is given by the Marchenko-Pastur distribution. We give a direct proof of this statement based on the recursion satisfied by the Laguerre-polynomials. At the same time, our main result gives that the leading term in $p$ and $(α+p)$ of the sum of the $k^{th}$ power of the roots of $L_{p}^{(α)}$ coincides with the $k^{th}$ moment of the Marchenko-Pastur law. We also mention the fact that the expectation of the characteristic polynomial of a $XX^T$ type random covariance matrix, where $X$ is a $p\times n$ random matrix with iid elements, is $\ell^{(n-p)}_p$, i.e. the monic version of the $p^{th}$ Laguerre polynomial with parameter $n-p$.

math.CA↗