Statistics of spectral properties of matrix-valued random variables.

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135 views

### A matrix rank problem over finite fields: Is that a known problem?

I have already asked the same question on cstheory.SE, but I haven't got an acceptable answer. So, I decided to ask it here.
It might be a known problem, however.
Let $A \odot B$ denote elementwise ...

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votes

**1**answer

169 views

### maximum of certain Gaussian processes

Let $\mathbf{a}_k\in\mathbb{C}^n$ for $k=1,2,\ldots,m$ be i.i.d. standard complex normal random vectors with distribution $c\mathcal{N}(0,\mathbf{I})$. I am interested in a tight upper bound on the ...

**2**

votes

**1**answer

75 views

### Spectral norm tail bound of a correlated random matrix

I am looking for the tail bound of spectral norm for certain type of random matrix.
Let's say we have a $n\times n$ symmetric random matrix $R$, and for each entry $R_{ij}$, we have that
$$
...

**7**

votes

**1**answer

146 views

### Distribution of entries of a doubly-sorted random matrix

Take an $n \times n$ random matrix whose entries are i.i.d. with uniform distribution in $[0,1]$. Look at the sums of the elements of each row and then permute the rows so that these sums form an ...

**7**

votes

**1**answer

204 views

### Concentration of sum of powers of normals

Let $Z_1,Z_2,\ldots,Z_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables ...

**18**

votes

**1**answer

1k views

### Unexpected $\sqrt{3}$

A somewhat lengthy calculation, involving integrals, reveals that the probability an $n\times n$ Hermitian matrix, drawn from the Gaussian unitary ensemble, is positive definite, decays as ...

**2**

votes

**0**answers

99 views

### Finite Volume 1D Anderson Tight Binding Model

My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on $\mathbb{Z}_n$ for $n \in \mathbb{N}$. For my question, these are the ...

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votes

**1**answer

79 views

### Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end.
$\mathrm{\underline{Background}}$
Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$
(with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...

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votes

**1**answer

66 views

### Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix

Let $Q$ be a random variable taking as its values the set of $n \times k$ real matrices with orthogonal columns, and whose distribution is the Haar measure on the Stiefel manifold $O(n)/O(n-k)$. This ...

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votes

**0**answers

53 views

### Compare full-rank probabilities of products of random matrices

Consider two matrices $C_1=A\times B_1$ and $C_2=A\times B_2$, where $A\in\mathbb{F}_q^{N\times K}$, $B_1\in\mathbb{F}_q^{K\times M}$ and $B_2\in\mathbb{F}_2^{K\times M}$; $M\leq N\leq K$.
It is ...

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vote

**0**answers

61 views

### integrality of a linear program — binary equality constaints

Consider the following linear program:
$\left\{
\begin{array}{l}
\underset{x}{max} \;\;c^Tx\\
[I, \;B]x = \mathbf{1}\\
x\geq 0
\end{array}
\right.$
where $c$ is a vector ...

**4**

votes

**1**answer

128 views

### Estimating the probability that $\|Av\| \ge \|v\|$

Given a diagonalizable matrix $A \in \mathbb{R}^{n \times n}$ with real eigenvalues, satisfying $1+c_1 \le \rho(A) \le 1+c_2$ $(0<c_1 \le c_2)$, obviously there exists a $v \in \mathbb{R}^{n}$ such ...

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votes

**0**answers

82 views

### Random square submatrices of a Hadamard matrix

Question: For $N$ be a power of $2$, let $A$ be a random $d \times d$ submatrix of the $N \times N$ Hadamard matrix (the matrix of the Hadamard/Walsh-Fourier transform). What is the best known upper ...

**3**

votes

**1**answer

129 views

### Is there any theoretical results about the determinants of a Non-Central Wishart matrix?

As we know that a Non-Central Wishart matrix is defined as
$W:=XX^T$, where $X \in \mathbb{R}^{p \times N}$, and
$X:= M + E$, with $M \in \mathbb{R}^{p \times N}$ a deterministic and non-zero matrix, ...

**5**

votes

**1**answer

173 views

### concentration of random matrices involving normal random variables

Define the random variable
\begin{align*}
A=|a_1|^2\mathbf{a}\mathbf{a}^*
\end{align*}
where $\mathbf{a}\in\mathbb{c}^n$ is a random vector distributed as ...

**3**

votes

**1**answer

165 views

### Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...

**11**

votes

**0**answers

268 views

### What kind of random matrices have rapidly decaying singular values?

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the ...

**1**

vote

**1**answer

77 views

### ordinary least square and random projection

Let $X$ a $d \times T$ given matrix and $M$ a $n \times d$ random matrix (say i.i.d. centered coefficients). Define $Y=MX$ in $\mathbb{R}^n$ and $H=Y'(YY')^{-1}Y$ where $'$ denotes the transpose ...

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votes

**1**answer

62 views

### Determine the expected size of a lower triangular sub-matrix of a random matrix?

Consider a $N\times N$ random matrix $A=[A_{ij}]$, whose elements are independently randomly chosen from the binary field $\mathbb{F}_2=\{0,1\}$ with probabilities $p_0=p$ and $p_1=1-p$. Suppose that ...

**5**

votes

**0**answers

169 views

### Kasteleyn, Gessel-Viennot and eigenvalues

The Kasteleyn matrix (for counting perfect matchings) and the Lindström-Gessel-Viennot matrix (for counting families of nonintersecting lattice paths) are tightly related, as observed many times by ...

**2**

votes

**1**answer

251 views

### Distribution of sum of freely independent Marchenko-Pastur measures

Given freely independent random variables $X_i$ with Marchenko-Pastur measures $\mu_i$, $i\in\{1,\dots,n\}$ how can we find the distribution of the scaled sum of these random variables ...

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vote

**1**answer

279 views

### Random matrix determinant problem

Suppose we have a a set of random matrices in the complex field of the form $a_iv_iv_i^H$ for $i=\{1,\dots,n\}$ where $a_i$ are constant positive real scalars and $v_i$ are random complex valued ...

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votes

**0**answers

519 views

### Probability a random Toeplitz matrix is singular

Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the ...

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votes

**1**answer

250 views

### Bounds on the eigenvalues of a random binary matrix

Consider $A$, a random binary matrix of zeros and ones in $\mathbb{R}^{{M\times N}}$, and $M>N$. We assume that $P(a_{i,j}=0)=P(a_{i,j}=1)=0.5$ (although I appreciate any advice on the case of ...

**0**

votes

**1**answer

326 views

### Null space of random $(0,1)$ binary matrix [closed]

What can be said about the null space of random $(0,1)$ rectangular binary matrices? In particular, I am interested in the probability that there is any non-zero vector with only integer coordinates ...

**0**

votes

**0**answers

82 views

### Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...

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votes

**0**answers

160 views

### Distributions of eigenvalues for matrix normal distribution: related references

I am interested in the distribution of the eigenvalues of matrices that are sampled from the matrix normal distribution.
I am sampling from $p(X \mid M,U,V)$ and let's assume that I know the ...

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votes

**1**answer

202 views

### Relation between the eigenvalue density and the resolvent?

Disclaimer: This is a cross-post from Math Underflow. Given that there is little activity on the subject (random-matrice) on the aformentioned site, and given that many interesting discussion on this ...

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votes

**1**answer

196 views

### “Ergodicity” for eigenvalues of random matrices?

Sorry if the wording of this question is sloppy, I have a weak background in probability theory (hence the quotation marks throughout).
Is there some "ergodicity-type" result for Wigner's semicircle ...

**2**

votes

**2**answers

173 views

### Probability for a random positive-semidefinite matrix to not be positive-definite?

If I take $A^TA$, where $A$ is a full-rank random matrix (let's say with Gaussian-distributed independent entries), can I expect it to be positive-definite? It will be positive semi-definite ...

**-1**

votes

**1**answer

241 views

### Rank of a random matrix

Let $x$ a random Gaussian vector of size $n$ with i.i.d coefficients $N(0,1)$. Let $J$ a random matrix with i.i.d coefficients $N(0,\sigma^2/n)$ where $\sigma \in [0,1]$. For any integer T>n, define:
...

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vote

**0**answers

114 views

### expected inverse of circulant plus random diagonal

I have a deterministic circulant matrix $R$ and a random diagonal matrix $X$ where all elements are IID and positive.
I need to determine the expected inverse of $R+X$, that is: Evaluate, in closed, ...

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votes

**0**answers

62 views

### Packing symmetric matrices in spectral norm, and defining measures on symmetric matrices

I'm trying to upper bound the $\epsilon$-packing number of $\Theta=\{A\in\mathbb{S}^{d}:\; a\preceq A \preceq b\}$ (where $\mathbb{S}$ are symmetric $d\times d$ matrices) for some $a\leq b$ with ...

**3**

votes

**2**answers

189 views

### Eigenvalue distribution of the sum of two random matrices

Suppose $D$ is a diagonal matrix of size $n \times n$ with diagonal elements $D_{ii}$ which are independent standard centered Gaussian random variables. Then consider a matrix $J$ such that its ...

**0**

votes

**0**answers

274 views

### expected matrix inverse of circulant plus diagonal matrix with chi-square variables

Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$.
Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...

**0**

votes

**1**answer

156 views

### concentration of sums of fourth moment of normals

I was wondering what is the best tail bound for
\begin{equation*}
\mathbb{P}\bigg\{\sum_{k=1}^n X_k^4>(1+t)3n\bigg\}\le ?
\end{equation*}
where $X_k$ are i.i.d. $\mathcal{N}(0,1)$.

**3**

votes

**1**answer

156 views

### Condition number of a random 0-1 matrix

Consider a 0-1 integer $n \times n$ matrix with coefficients chosen uniformly over $\{0,1\}$. The probability that it is singular is exponentially small, and so we expect that it has a well-defined ...

**15**

votes

**0**answers

293 views

### Quasi-classical limit of representation theory

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...

**1**

vote

**1**answer

140 views

### Determining the asymptotic behavior of some function of random matrix

Consider a series of random matrices $X_n\in\mathbb{R}^{n\times m}$ consisting of i.i.d. entries, each with zero mean and variance $1/m$, and let $a_n,b_n\in\mathbb{R}^{n\times1}$ be two deterministic ...

**5**

votes

**0**answers

134 views

### Behavior of eigenspaces of adjacency matrices of random graphs (not via perturbation theory)

For the sake of discussion, let us say that we have the adjacency matrix $A$ of a graph, on $n$ nodes, from a stochastic block model with 2 blocks. Another name for this (usually used in computer ...

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votes

**1**answer

136 views

### How to calculate eigenvalue density function of $XX^\dagger$ from the density function of X

Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary ...

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votes

**2**answers

606 views

### Laws of Iterated Logarithm for Random Matrices and Random Permutation

The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then
$$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, ...

**0**

votes

**1**answer

86 views

### Can I obtain the limit value of a linear spectral statistics using Stieltjes transform?

I would like to calculate the limit value of a linear functional
\begin{equation}
\lim_{n\rightarrow\infty}\mathcal{I}_n=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n ...

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votes

**1**answer

120 views

### What is the spectral norm of a random projection times a diagonal?

Take $n\ll N$. Let $P$ be an $n\times N$ matrix of iid $\mathcal{N}(0,1)$ random variables, and let $D$ be an $N\times N$ diagonal matrix.
What can be said about the distribution of the largest ...

**2**

votes

**1**answer

193 views

### Determining the asymptotic behavior of random matrices with vanishing ratio dimensions

Consider an $N\times K$ random matrix $X$ (defined on a probability space $(Ω,F,μ)$) with i.i.d. entries having zero mean and variance $1/K$.
There are a lot of results regarding the asymptotic ...

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votes

**1**answer

115 views

### GOE Version of Longest Increasing Subsequence

Let $S_n$ be the symmetric group equipped with uniform measure. For any $\pi\in S_n$, let $L_n=L_n(\pi)$ denote the longest increasing subsequence. A celebrated result of Baik, Deift and Johansson ...

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votes

**1**answer

192 views

### Tail bound for $L_2$ norm of top $k$ singular values of a random matrix

Let $Y=X^\top W$ , with $X, W \in \mathbb{R}^{d \times d}$ are random matrices with standard normal entries. Let $\lambda_j$ be the $j^{th}$ singular value of $Y$. Is there a way to bound the tail ...

**5**

votes

**1**answer

294 views

### Lower bound of integral involving Laguerre polynomials

I want to lower bound the expected value of the square root of a randomly chosen eigenvalue of a Wishart matrix.
To get the bound I want I need a lower bound on
$$T_n = ...

**4**

votes

**1**answer

235 views

### Measure concentration for weakly dependent random variables

For an application quite alien to probability theory, I'd like to have a kind of measure concentration estimate, in the following spirit. Suppose that to every $1\le i,j\le n$ there corresponds a ...

**10**

votes

**1**answer

362 views

### A simple proof for a theorem of Szekeres and Turán

Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...