اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدPerturbation of linear forms of singular vectors under Gaussian noise
90
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $Ainmathbb{R}^{mtimes n}$ be a matrix of rank $r$ with singular value decomposition (SVD) $A=sum_{k=1}^rsigma_k (u_kotimes v_k),$ where ${sigma_k, k=1,ldots,r}$ are singular values of $A$ (arranged in a non-increasing order) and $u_kin {mathbb R}^m, v_kin {mathbb R}^n, k=1,ldots, r$ are the corresponding left and right orthonormal singular vectors. Let $tilde{A}=A+X$ be a noisy observation of $A,$ where $Xinmathbb{R}^{mtimes n}$ is a random matrix with i.i.d. Gaussian entries, $X_{ij}simmathcal{N}(0,tau^2),$ and consider its SVD $tilde{A}=sum_{k=1}^{mwedge n}tilde{sigma}_k(tilde{u}_kotimestilde{v}_k)$ with singular values $tilde{sigma}_1geqldotsgeqtilde{sigma}_{mwedge n}$ and singular vectors $tilde{u}_k,tilde{v}_k,k=1,ldots, mwedge n.$ The goal of this paper is to develop sharp concentration bounds for linear forms $langle tilde u_k,xrangle, xin {mathbb R}^m$ and $langle tilde v_k,yrangle, yin {mathbb R}^n$ of the perturbed (empirical) singular vectors in the case when the singular values of $A$ are distinct and, more generally, concentration bounds for bilinear forms of projection operators associated with SVD. In particular, the results imply upper bounds of the order $Obiggl(sqrt{frac{log(m+n)}{mvee n}}biggr)$ (holding with a high probability) on $$max_{1leq ileq m}big|big<tilde{u}_k-sqrt{1+b_k}u_k,e_i^mbig>big| {rm and} max_{1leq jleq n}big|big<tilde{v}_k-sqrt{1+b_k}v_k,e_j^nbig>big|,$$ where $b_k$ are properly chosen constants characterizing the bias of empirical singular vectors $tilde u_k, tilde v_k$ and ${e_i^m,i=1,ldots,m}, {e_j^n,j=1,ldots,n}$ are the canonical bases of $mathbb{R}^m, {mathbb R}^n,$ respectively.
قيم البحث
اقرأ أيضاً
We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some interesting cons
equences. Our second result gives a characterization of limits in law for sequences of such vectors.
We study the question of when a ({0,1})-valued threshold process associated to a mean zero Gaussian or a symmetric stable vector corresponds to a {it divide and color (DC) process}. This means that the process corresponding to fixing a threshold leve
l $h$ and letting a 1 correspond to the variable being larger than $h$ arises from a random partition of the index set followed by coloring {it all} elements in each partition element 1 or 0 with probabilities $p$ and $1-p$, independently for different partition elements. While it turns out that all discrete Gaussian free fields yield a DC process when the threshold is zero, for general $n$-dimensional mean zero, variance one Gaussian vectors with nonnegative covariances, this is true in general when $n=3$ but is false for $n=4$. The behavior is quite different depending on whether the threshold level $h$ is zero or not and we show that there is no general monotonicity in $h$ in either direction. We also show that all constant variance discrete Gaussian free fields with a finite number of variables yield DC processes for large thresholds. In the stable case, for the simplest nontrivial symmetric stable vector with three variables, we obtain a phase transition in the stability exponent $alpha$ at the surprising value of $1/2$; if the index of stability is larger than $1/2$, then the process yields a DC process for large $h$ while if the index of stability is smaller than $1/2$, then this is not the case.
We consider the asymptotic behavior of the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes converges in distribution to a generalized Ornstein-Uhlenbec
k process. Our result considerably extends classical results to singular kernels, including the Biot-Savart law. The result applies to the point vortex model approximating the 2D incompressible Navier-Stokes equation and the 2D Euler equation. We also obtain Gaussianity and optimal regularity of the limiting Ornstein-Uhlenbeck process. The method relies on the martingale approach and the Donsker-Varadhan variational formula, which transfers the uniform estimate to some exponential integrals. Estimation of those exponential integrals follows by cancellations and combinatorics techniques and is of the type of large deviation principle.
A differential 1-form $alpha$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $alpha wedge (dalpha)^n eq 0$ almost everywhere, but which vanishes at a point $O$, i.e. $alpha (O) = 0$, is called a textit{singular contact
form} at $O$. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form $omega$ in dimension $2n$, i.e. differential 1-forms $gamma$ which vanish at a point and such that $dgamma = omega$, and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin cite{Lychagin-1stOrder1975}, Webster cite{Webster-1stOrder1987} and Zhitomirskii cite{Zhito-1Form1986,Zhito-1Form1992}. We make use of both the classical normalization techniques and the toric approach to the normalization problem for dynamical systems cite{Zung_Birkhoff2005, Zung_Integrable2016, Zung_AA2018}.
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 another fixed number.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
