Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userScaling positive random matrices: concentration and asymptotic convergence
76
0
0.0
(
0
)
Added by
Boris Landa
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Authors
Boris Landa
Ask ChatGPT about the research
No Arabic abstract
It is well known that any positive matrix can be scaled to have prescribed row and column sums by multiplying its rows and columns by certain positive scaling factors (which are unique up to a positive scalar). This procedure is known as matrix scaling, and has found numerous applications in operations research, economics, image processing, and machine learning. In this work, we investigate the behavior of the scaling factors and the resulting scaled matrix when the matrix to be scaled is random. Specifically, letting $widetilde{A}inmathbb{R}^{Mtimes N}$ be a positive and bounded random matrix whose entries assume a certain type of independence, we provide a concentration inequality for the scaling factors of $widetilde{A}$ around those of $A = mathbb{E}[widetilde{A}]$. This result is employed to bound the convergence rate of the scaling factors of $widetilde{A}$ to those of $A$, as well as the concentration of the scaled version of $widetilde{A}$ around the scaled version of $A$ in operator norm, as $M,Nrightarrowinfty$. When the entries of $widetilde{A}$ are independent, $M=N$, and all prescribed row and column sums are $1$ (i.e., doubly-stochastic matrix scaling), both of the previously-mentioned bounds are $mathcal{O}(sqrt{log N / N})$ with high probability. We demonstrate our results in several simulations.
rate research
Read More
The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spectral measures in all cases.
The Hartman-Watson distribution with density $f_r(t)$ is a probability distribution defined on $t geq 0$ which appears in several problems of applied probability. The density of this distribution is expressed in terms of an integral $theta(r,t)$ which is difficult to evaluate numerically for small $tto 0$. Using saddle point methods, we obtain the first two terms of the $tto 0$ expansion of $theta(rho/t,t)$ at fixed $rho >0$. An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in $rho$. As an application we obtain the leading asymptotics of the density of the time average of the geometric Brownian motion as $tto 0$. This has the form $mathbb{P}(frac{1}{t} int_0^t e^{2(B_s+mu s)} ds in da) = (2pi t)^{-1/2} g(a,mu) e^{-frac{1}{t} J(a)} (1 + O(t))$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory.
In this article, we consider the preconditioned Hamiltonian Monte Carlo (pHMC) algorithm defined directly on an infinite-dimensional Hilbert space. In this context, and under a condition reminiscent of strong log-concavity of the target measure, we prove convergence bounds for adjusted pHMC in the standard 1-Wasserstein distance. The arguments rely on a synchronous coupling of two copies of pHMC, which is controlled by adapting elements from arXiv:1805.00452.
Let $A$ and $B$ be two $N$ by $N$ deterministic Hermitian matrices and let $U$ be an $N$ by $N$ Haar distributed unitary matrix. It is well known that the spectral distribution of the sum $H=A+UBU^*$ converges weakly to the free additive convolution of the spectral distributions of $A$ and $B$, as $N$ tends to infinity. We establish the optimal convergence rate ${frac{1}{N}}$ in the bulk of the spectrum.
We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
