Statistics of spectral properties of matrix-valued random variables.

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3
votes
1answer
117 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
1answer
172 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
1answer
162 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
0answers
259 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
1answer
74 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 ...
0
votes
1answer
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
0answers
157 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
1answer
241 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 ...
1
vote
1answer
259 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 ...
11
votes
0answers
510 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 ...
0
votes
1answer
231 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
1answer
307 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
0answers
79 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 ...
2
votes
0answers
147 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 ...
0
votes
1answer
183 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 ...
5
votes
1answer
193 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
2answers
169 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
1answer
233 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: ...
1
vote
0answers
113 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, ...
2
votes
0answers
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
2answers
187 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
0answers
264 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
1answer
152 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
1answer
150 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
0answers
285 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
1answer
139 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 ...
4
votes
0answers
121 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 ...
0
votes
1answer
133 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 ...
14
votes
2answers
599 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
1answer
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 ...
2
votes
1answer
114 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
1answer
190 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 ...
8
votes
1answer
113 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 ...
2
votes
1answer
191 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
1answer
287 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
1answer
229 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
1answer
361 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 ...
14
votes
2answers
390 views

Repeated random two-steps in $\mathbb{R}^3$: unbounded?

I created a random isometry $T$ of $\mathbb{R}^3$ by generating a random orthogonal matrix $M$, uniformly distributed among all such, and a random displacement $v$, whose coordinates are drawn from a ...
9
votes
1answer
276 views

Complexity of the union of randomly rotated unit cubes

It is a remarkable fact that the union of congrent cubes has only at most near-quadratic combinatorial complexity, $O^*(n^2)$ for $n$ cubes, known to be almost tight. This contrasts with the union of ...
5
votes
0answers
178 views

Quasicompactness of transfer operators associated to IID matrix products

Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
0
votes
1answer
237 views

Determine the probability that two random vectors over a finite field are orthogonal

Hi all, Suppose that $\mathbf{f}=[f_1, f_2,\ldots,f_m]$ and $\mathbf{g}=[g_1,g_2,\ldots,g_m]$ are two $m$-dimensional vectors. All $f_i$'s are chosen uniformly randomly from a finite field ...
2
votes
0answers
129 views

Error bound on matrix vector multiplication

I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate. Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. ...
0
votes
1answer
126 views

Find a generalized hypergeometric-based function yielding certain ratios of fifth-degree polynomials

Find a (presumably, generalized hypergeometric-based function $f(n,a,k)$), yielding for $n=1, a=\frac{1}{2}$,the rational function (ratio of fifth-degree polynomials) \begin{equation} ...
3
votes
2answers
141 views

Non-asymptotic results for bulk of random Wishart matrix

Let $n$ be a positive integer and let $X_n$ be an $n\times n$ random matrix whose entries are iid standard gaussian random variables. I am interested in the distribution of the average singular value ...
9
votes
2answers
637 views

Probability of random (0,1) Toeplitz matrix being invertible

A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant. What is the probability that a random $n \times n$ binary Toeplitz ...
0
votes
0answers
237 views

Notation for a functional L2 matrix norm

Hi, Let $v(z)$ be a 2x2 matrix depending on a complex variable z, defined on an oriented contour $\Sigma$ in the complex plane. Can anyone tell me the meaning of the notation: ...
10
votes
0answers
173 views

What are the difficulties in proving almost-everywhere stability of Gaussian elimination?

It is well known that Gaussian elimination without pivoting is numerically unstable, and in practice Gaussian elimination is done with row pivoting (partial pivoting). A theorem of Wilkinson states ...
8
votes
2answers
785 views

Intuition behind the spectral density of random matrices

Hi, I have read that the spectral density of an NxN random matrix consisting of iid random variables with zero mean and unit variance converges as N goes to infinity to the uniform distribution on ...
0
votes
1answer
356 views

Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...
3
votes
0answers
174 views

Matrix where every subset of rows has maximal rank

I am looking for a class of matrices $M(n(m), m, k(m), \phi)$ with the following properties: M is $n \times m$ where $n(m) > m$. Every subset of rows of size $k$ has (maximal) rank $m$. $n(m)$ ...