Entropic CLT and phase transition in high-dimensional Wishart matrices

@article{Bubeck2015EntropicCA,
title={Entropic CLT and phase transition in high-dimensional Wishart matrices},
author={S{\'e}bastien Bubeck and Shirshendu Ganguly},
journal={ArXiv},
year={2015},
volume={abs/1509.03258}
}
• Published 10 September 2015
• Computer Science, Mathematics
• ArXiv
We consider high dimensional Wishart matrices $\mathbb{X} \mathbb{X}^{\top}$ where the entries of $\mathbb{X} \in {\mathbb{R}^{n \times d}}$ are i.i.d. from a log-concave distribution. We prove an information theoretic phase transition: such matrices are close in total variation distance to the corresponding Gaussian ensemble if and only if $d$ is much larger than $n^3$. Our proof is entropy-based, making use of the chain rule for relative entropy along with the recursive structure in the…
Asymptotic Behavior of Large Gaussian Correlated Wishart Matrices
• Mathematics, Computer Science
Journal of Theoretical Probability
• 2021
The situation where the row-independence assumption is relaxed is analyzed and the setting of random $p$-tensors ($p\geq 3$), a natural extension of Wishart matrices, is analyzed.
Limiting behavior of large correlated Wishart matrices with chaotic entries
• Mathematics
• 2020
We study the fluctuations, as $d,n\to \infty$, of the Wishart matrix $\mathcal{W}_{n,d}= \frac{1}{d} \mathcal{X}_{n,d} \mathcal{X}_{n,d}^{T}$ associated to a $n\times d$ random matrix
The middle-scale asymptotics of Wishart matrices
• Mathematics
The Annals of Statistics
• 2019
We study the behavior of a real $p$-dimensional Wishart random matrix with $n$ degrees of freedom when $n,p\rightarrow\infty$ but $p/n\rightarrow 0$. We establish the existence of phase transitions
A CLT in Stein’s Distance for Generalized Wishart Matrices and Higher-Order Tensors
We study the convergence along the central limit theorem for sums of independent tensor powers, $\frac{1}{\sqrt{d}}\sum\limits_{i=1}^d X_i^{\otimes p}$. We focus on the high-dimensional regime where
A Smooth Transition from Wishart to GOE
• Mathematics
• 2016
It is well known that an $$n \times n$$n×n Wishart matrix with d degrees of freedom is close to the appropriately centered and scaled Gaussian orthogonal ensemble (GOE) if d is large enough. Recent
A high-dimensional CLT in $$\mathcal {W}_2$$W2 distance with near optimal convergence rate
The main feature of the theorem is that the rate of convergence is within\log n$$logn of optimal for n, d \rightarrow \infty$$n,d→∞.
Phase transition in noisy high-dimensional random geometric graphs
• Mathematics, Computer Science
ArXiv
• 2021
The soft high-dimensional random geometric graph G(n, p, d, q), where each of the n vertices corresponds to an independent random point distributed uniformly on the sphere S, is considered, and the probability that two vertices are connected by an edge is a decreasing function of the Euclidean distance between the points.
High-dimensional regimes of non-stationary Gaussian correlated Wishart matrices
• Mathematics, Computer Science
Random Matrices: Theory and Applications
• 2020
It is shown that the convergences stated above also hold in a functional setting, namely as weak convergence in [Formula: see text], and it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law.
A probabilistic view of latent space graphs and phase transitions
• Mathematics, Computer Science
ArXiv
• 2021
This work considers a one-parameter family of random graphs, characterized by the variance of this random variable, that smoothly interpolates between a random dot product graph and an Erdős–Rényi random graph, and proves phase transitions of detecting geometry in these graphs, in terms of the dimension of the underlying geometric space and the variance parameter of the conditional probability.
New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems
• Mathematics
The Annals of Applied Probability
• 2022
We extend Stein's celebrated Wasserstein bound for normal approximation via exchangeable pairs to the multi-dimensional setting. As an intermediate step, we exploit the symmetry of exchangeable pairs

References

SHOWING 1-10 OF 29 REFERENCES
Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations
• Mathematics
• 2013
Let $$\lambda _1, \ldots , \lambda _n$$λ1,…,λn be random eigenvalues coming from the beta-Laguerre ensemble with parameter $$p$$p, which is a generalization of the real, complex and quaternion
Testing for high‐dimensional geometry in random graphs
• Computer Science, Mathematics
Random Struct. Algorithms
• 2016
The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix and a conjecture for the optimal detection boundary is made.
Entropy jumps for isotropic log-concave random vectors and spectral gap
• Mathematics
• 2012
We prove a quantitative dimension-free bound in the Shannon{Stam en- tropy inequality for the convolution of two log-concave distributions in dimension d in terms of the spectral gap of the density.
Concentration of mass on convex bodies
Abstract.We establish sharp concentration of mass inequality for isotropic convex bodies: there exists an absolute constant c >  0 such that if K is an isotropic convex body in $$\mathbb{R}^{n}$$,
Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles
• Mathematics
• 2009
Let K be an isotropic convex body in Rn. Given e > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to e with
Elements of Information Theory
• Computer Science
• 1991
The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Entropy jumps in the presence of a spectral gap
• Mathematics
• 2003
It is shown that if X is a random variable whose density satisfies a Poincare inequality, and Y is an independent copy of X, then the entropy of (X + Y )/ p 2 is greater than that of X by a fixed
MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS
• Mathematics
• 2007
Since the introduction of Stein's method in the early 1970s, much research has been done in extending and strengthening it; however, there does not exist a version of Stein's original method of
On the distribution of the largest eigenvalue in principal components analysis
Let x (1) denote the square of the largest singular value of an n x p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component