Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
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Bounded density for determinant of GOE
Let $M$ a random GOE matrix, i.e. $M=(M_{i,j})$ is a symmetric matrix and the $M_{i,j},i\leq j$ are independent centred Gaussien entries with variance 1, except on the diagonal where the variance is $...
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Given iid $w_1,\ldots,w_N \sim N(0,1/d)$ iid, find a simple matrix $A$ s.t $\|aa^T-A\|_{op} \to 0$, where $a_i := E_{G \sim N(0,1)}[f(\|w_i\| G)]$
Let $d$ and $N$ be two large comparable integers, for example assume
$$
N,d \to \infty, \quad d/N \to \gamma \in (0,\infty).
$$
Let $w_1,\ldots,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \...
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Maximum column norm of random $A^{-1}B$
Suppose that $A$ is an $n$ by $n$ Gaussian matrix (each component i.i.d. normal distributed with mean 0 and variance 1). Let $b$ be a $n$-Gaussian vector. Then it could be easily proven that the ...
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How to analyze the value of convergence of functions of random matrices?
Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of ...
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Multilinear non-commutative Khintchine inequality
Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...
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Computing Haar measure of matrices sampled from SO(n)
I am looking to sample uniform matrices from SO(n).
I know that uniform matrices can be sampled from O(n) by taking the QR decomposition of Gaussian random square matrices and adjusting the sign of ...
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Distribution of the constraint matrix conditioned on the solution of the linear system
Suppose that A is a random matrix in $R^{n\times n}$, with each component independently and identically distributed (iid) according to $\mathcal{N}(0,1)$. Additionally, b is a random vector in $R^n$, ...
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A function with unexpectedly simple Legendre transformation
Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and
\begin{equation}
I(x)=
\begin{cases}
\frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\
\...
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Question about the spectrum of a deformed GOE matrix
Consider a fixed real value $\sigma>0$. Let $A,Z$ be two independent $n\times n$ GOE matrices, and define $B=A+\sigma Z$. I am interested in finding a bound (possibly dependent on $n$) for the ...
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Convergence of edge eigenvalues for Gaussian matrices
I am reading this lecture note.
I have a difficulty in understanding the third section in chapter 6. Particularly, in Theorem 4.1, they claimed that
Let $X$ be a Gaussian Wigner matrix satisfying ...
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High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)
Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
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Concentration of the norm of subGaussian random vectors
I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin.
I have a sub-Gaussian vector $y$, in $\mathbb{R}^n$ and sub-Gaussian norm $C$ non dependent on $n$. I ...
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Eigenvalue distribution of a band matrix
Let $\mathbf M_i$ be rectangular matrices of dimensions $N_{i-1}\times N_i$. We assume that their entries are random, with zero mean and variance $\sigma_i^2$.
For some positive integer $k$, I define ...
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Functional calculation for Hermitian matrices
First, let me recall some useful definitions
We recall that if $A=U \Lambda U^*$ is a Hermitian matrix with $U U^*=U^* U=I$ and $\Lambda=\operatorname{diag}\left(\lambda_1\right)$ and $f: \mathbb{R} \...
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Convergence in probability of quadratic form with positive mean
Let $\boldsymbol{X}_n\in\mathbb{R}^n$ be a sequence of Gaussian random vectors with independent entries, such that $X_{n,i}\sim \mathcal{N}(\mu_i,\sigma^2)$ (that is, all entries of the $n$th vector ...
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Is there a closed-form solution for $\max_D \operatorname{Tr}(ADBD)$
Is there a closed-form solution for
$$\max_D \operatorname{Tr}(ADBD)$$
where $D$ is a $N\times N$ diagonal matrix with $m<N$ number of $1$'s and the rest are $0$'s, and $A$ and $B$ are real ...
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Limiting value of trace of resolvent matrix involving two independent Wishart random matrices
Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
d/n_k \to \phi_k \in (0,\infty).
$$
Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
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Computation of Brown measure of the shift operator on $\ell^2(\mathbb N)$?
This looks an extremely simple question - I am just trying to give an example of Brown measure, https://en.wikipedia.org/wiki/Brown_measure, so I try to compute it for the left/right-shift operator on ...
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Concentration inequality for minimal eigenvalue of sample covariance
I was reading an article of matrix completion and met the following lemma
The concentration inequality for $\sigma_{\max}$ part is a standard result. However, I didn't find any results like the $\...
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matrix bernstein's inequality: from tail probability to expectation
Let $X_i$ be independent, mean zero, $n\times n$, symmetric random matrices. $\|X_i\|\leq K$ almost sure for $\forall I$.
We have matrix Bernstein's inequality for the tail probability as follows
$$\...
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Maximizing the trace of the resolvent of a Wishart matrix over positive unit trace matrices?
Let $G$ be a standard $d \times d$ Wishart random matrix and consider the problem of maximizing the function
$$
f(M) = \mathbb{E}\Big[\mathrm{tr}((G + M^{-1})^{-1})\Big],
$$
over the class of real ...
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Eigenvalue analysis of $X^T (XX^T + \mathrm{Id})^{-1} X$ for $X$ iid random matrix
Consider the following quantity
$$X^T (XX^T + \mathrm{Id})^{-1} X,$$
where $X \in \mathbb{R}^{m\times n}$ is a iid random matrix with 0 mean and finite variance.
The empiric covariance matrix ${X^T X}$...
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Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$
Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5
$$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$
...
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Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)
Let $g:\mathbb R \to \mathbb R $ be a continuous function which is
"sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and
"sufficiently integrable" (e.g integrable w.r.t $N(0,...
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A second-order recursion (functional equation)
In a calculation of some momenta of random matrices (GOE), I encounter a functional equation, in the form of a second-order recursion,
$$L(s+1)=L(s)+2s(2s+1)L(s-1).$$
Is it familiar to someone ? Is ...
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Asymptotic limit of trace of random matrix $(aI_m + WW^\top)^{-1}$, where $W$ has iid rows from $N(0,\Sigma)$
Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a ...
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What are applications of asymptotic freeness of random matrices?
In around 1990 Voiculescu showed asymptotic freeness of certain random matrices,
i.e., free independence when the matrix size goes to infinity.
Since then this link between free probability and random ...
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Conditional distributions of random orthogonal projection matrix
I have encountered a rather curious question.
Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n$-dimensional ...
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Moment method / genus expansion for random matrices with i.i.d. entries
Given a (say real) random matrix $M=(M_{i,j})_{1\leq i, j \leq N}$, the moments method consists in computing (the limits in $N$ of) the quantities $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$
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Expectation of the trace of inverse of a Gaussian random matrix
Given a $N×M$ random complex gaussian matrix $X$ and $N×K$ random complex gaussian matrix $Y$ I'm interested in approximating the expectation expressed as:
\begin{align}
E[trace({(aX{X^H} + I)^{ - ...
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The problems of global asymptotic freeness
Let $X_{N}\in\mathcal{M}_{N}\big(L^{\infty-}(\Omega,\mathbb{P})\big)$ be a $N\times N$ random complex matrix such its entries $(x_{ij}, 1\leq i, j\leq N)$ be $i.i.d.$, centred with variance $1$. $X_{...
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Distribution of inverse of a random matrix
I got stuck into a problem and couldn't find its
satisfactory answer anywhere.
My question is simple. Suppose I have a fat random matrix (i,e., $R$ has dimensions $k\times d$ where $k<d$) whose
...
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$\log\det$ asymptotics of a skew-circulant matrix with additive diagonal bimodal disorder
I'd like to share a problem that I have been dealing with for a longer time now.
In the framework of quenched disorder in the square-lattice Ising model I want to calculate, for large even $M$, the ...
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Dimension-free sample complexity for the inverse of Gaussian sample covariance?
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in ...
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Concentration inequality for norm of solution to nonlinear least-squares problem
Define the piecewise-linear function $\psi(t):=\max(t,0)$ for all $t \in \mathbb R$.
Let $d,n,k \to \infty$ at the same rate (i.e $n \asymp k \asymp d$).
Let $y_1,\ldots,y_n \in \{-1,1\}$ uniformly ...
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How to derive the mean of inverse-Wishart distribution?
How to derive the mean of inverse-Wishart distribution in
Inverse-Wishart distribution?
I have no idea about it. Thanks for your help.
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Quantifying the amount of structure in a data set via random matrix theory
Given a data matrix, $M \in \mathbb{R}^{n \times p}$, I am interested in methods quantifying the amount of structure in present in $M$.
I've found a few approaches, but I would like to learn more ...
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Minimizing the largest eigenvalue of random matrices
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix with entries $A_{ij} \sim \mathcal{N} (0,1)$, all independent except for the symmetry condition.
Consider the following minimization problem:...
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Random matrices: Relation between leading eigenvector and a vector in culumn space
Let $X$ be a $n\times n$ symmetric matrix with iid zero-mean random entries on and above the diagonal. Denote by $v$ the eigenvector corresponding to the largest eigenvalue of $X$. Let $a$ be a fixed $...
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The lattice spanned by $m$ random 0-1 vectors of length $n$
Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More ...
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What can we say about $\mathbb{E}[\mathrm{tr}A^{1/2}]$ for $A=\frac{1}{C}\sum_{i=1}^\infty c_i \alpha_i\alpha_i^\top \in\mathbb{R}^{m\times m}$?
Suppose we are given a summable sequence $(c_i)_{i\in\mathbb{N}}$ with $\sum_{i=1}^\infty c_i = C<\infty$ and independent $m$-dimensional, standard Gaussian vectors $\alpha_i\sim\mathcal{N}(0,I_m)$,...
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Maximum norm within a random subspace intersected with an ellipsoid
Let $d < n$, and let $G_n(d)$ denote the space of all $d$-dimensional subspaces of $\mathbb{R}^n$.
Let $a = (a_1,\dots, a_n)$ denote a positive sequence, and define
$U(a) = \{u \in \mathbb{R}^n: \...
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Expected value of MGIG distribution
I'm currently dealing with a Gibbs sampler of the multivariate generalized inverse Gaussian distribution (MGIG). In order to check the correctness of the sampler, I'd like to know the expected value ...
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Reference book on Riemann zeta function and random matrices
What is a reference book to understand the relation between the Riemann zeta function and random matrices?
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Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation
$\DeclareMathOperator\GL{GL}$A few months ago, the following discussion took place on AoPS, concerning matrices that have nonzero determinant under any permutation of their entries: https://...
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Expected norm of a product of Gaussian matrices
Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof?
$$E[\|C_n\|_F^2]=d^{n+1}$$
This ...
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Distribution of joint Gaussian conditional on their sum of squares
Given a random gaussian matrix $\mathbf{X}$ with zero mean matrix and covariance matrix $\mathbf{\Sigma}$, and two deterministic matrices $\mathbf{A}$ and $\mathbf{B}$. If I know the value of $\|\...
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Isolated eigenvalues of a random matrix
This is a continuation of this question. Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I want to study the eigenvalues of the matrix $O+O^\top + \lambda uu^\top$ where ...
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Expected value of the largest singular value of a random matrix with entries in $N (0,1)$
Given a matrix $A \in \mathbb R^{n \times n}$ whose entries are i.i.d. $N(0,1)$, what is the expected value of its largest singular value? Equivalently, what is the expected value of the largest ...
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Spectral density of symmetrized Haar matrix
Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I found by simulations that the spectral density of $O+O^\top$ is the arcsin law rescaled to the interval $[-2,2]$. I can'...
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