All Questions
74 questions
2
votes
2
answers
215
views
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 ...
CommunityBot
- 1
12
votes
1
answer
628
views
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 \\
\...
CommunityBot
- 1
2
votes
1
answer
1k
views
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 ...
CommunityBot
- 1
15
votes
2
answers
6k
views
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
...
CommunityBot
- 1
1
vote
1
answer
69
views
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 ...
CommunityBot
- 1
3
votes
0
answers
131
views
Matrix-Gaussian distributions
The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
- 193
3
votes
2
answers
512
views
Fourier transform of eigenvalue distribution of GUE matrices
I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $X\sim \mathrm{GUE} (d)$, ...
- 188k
1
vote
1
answer
99
views
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 ...
- 3,741
1
vote
0
answers
67
views
Random matrix theory: accounting for mean
Assume a random matrix, denoted as $X$, which is an $n$ by $T$ matrix, $T\geq n$. While I understand the typical scenario where the random variables $X_{ij}$ are sampled from a $\mathcal{N}(0,\sigma_{...
- 6,367
3
votes
1
answer
269
views
Trace of product of two Wishart matrices
Let $A,B$ be two independent complex Wishart matrices, $A,B\sim CW_p(\mathbf{I},n)$, that is $A=\frac1n GG^\dagger$& $B=\frac1n QQ^\dagger$ where $G$ and $Q$ are independent $p\times n$ complex ...
- 188k
6
votes
1
answer
894
views
Expected value of orthogonal projection $X^{+}X$
Let $X\in\mathbb{R}^{m\times n}$, where $m<n$, be a random matrix where the rows $x_i$ ($i=1,...,m$) are sampled i.i.d. from Gaussian distribution with mean $0$ and covariance $\Sigma$, i.e. $x_i\...
- 1
1
vote
1
answer
217
views
How to normalize an Inverse Wishart random matrix?
Background:
Let $d\in \mathbb{N}$.
Define the space of (real symmetric) positive definite matrices of size $d\times d$ as follows:
\begin{align}
\mathcal{S}_{++}^d := \big\{\mathbb{M}\in \mathbb{R}^{d\...
CommunityBot
- 1
3
votes
0
answers
334
views
Tail bound on trace norm / nuclear norm / Schatten-1 norm of Rademacher matrix
Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...
- 223
1
vote
1
answer
122
views
Expectation value of random GUE matrix
Let $A$ be a matrix of the Gaussian unitary ensemble (GUE) and $v_1,v_2$ be two orthonormal vectors.
I wonder if one can compute (or at least get a non-trivial lower bound on) the expectation value
$$\...
- 188k
1
vote
0
answers
50
views
Characterizing set of IID average of symmetric positive semidefinite matrices matrices
Let $\mathcal{S}_+^d$ denote the family of real $d \times d$ symmetric (strictly) positive definite matrices.
Define $\mathcal{P}_d$ to be those measures $\nu$ on $\mathcal{S}_+^d$ (assumed to have ...
- 400
2
votes
1
answer
185
views
Limiting distribution of "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$
Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.
...
1
vote
1
answer
157
views
Moments of rescaled Bernoulli random matrix
Suppose $X \in \{0,1\}^{n \times m}$ is a matrix generated according to the following generative process:
$$Z_{ij} \sim \text{Bernoulli}(p) \implies X_{ij} = \frac{Z_{ij}}{\sum_{k=1}^m Z_{ik}}.$$
Is ...
- 128k
1
vote
1
answer
226
views
Orthogonal transformation of multivariate Bernoulli-Gaussian distribution
Actually, I have asked this question in https://math.stackexchange.com/questions/4330127/orthogonal-transformation-of-multivariate-bernoulli-gaussian-distribution, but I think mathoverflow might be ...
- 188k
1
vote
0
answers
100
views
Exponential decay of a random matrix falling into a ball
Let $A=U\Sigma V^T\in\mathbb{R}^{n\times n}$ be a random matrix defined in the following way: $U,V$ are uniformly distributed on the orthogonal group $O(n)$, $\Sigma$ is a diagonal matrix such that ...
- 1,495
5
votes
0
answers
239
views
Expected value of $X^{\top}(XAX^{\top})^{-1}X$ for large random $X$
Let $X\in \mathbb{R}^{m\times n}$ be a random matrix where the entries are i.i.d. standard normal, and let $A\in \mathbb{R}^{n\times n}$ be a deterministic diagonal matrix with positive entries on the ...
- 188k
4
votes
0
answers
75
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1}
$...
0
votes
1
answer
209
views
Factorisation of Gaussian random matrix into random Hermitian and correction factor
By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries
$$\mathbf{\Gamma}_{n\times k}...
CommunityBot
- 1
1
vote
0
answers
225
views
Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
- 63.9k
0
votes
0
answers
156
views
Total variation convergence of random matrices and convergence of empirical spectral distributions
In the paper https://arxiv.org/pdf/1411.5713.pdf, on page 17, the authors prove in Theorem 7 that the total variation distance between the joint distribution of the entries of certain Wishart matrices ...
- 1
0
votes
0
answers
92
views
Linear independence of Wishart matrices
Let $W\sim W_n(I,d)$ be a real Wishart matrix of an identity covariance matrix and $d$ degrees of freedom, i.e., $W=XX^T$ for $X$ being an $n\times d$ matrix whose entries are i.i.d sampled from a ...
- 123
2
votes
1
answer
904
views
Diagonalizability of Gaussian random matrices
Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a ...
- 188k
0
votes
1
answer
1k
views
Expectation of inverse of random matrices
Assume that $\mathbf{X}$ is a random positive-definite matrix. Then, is there any upper or lower bound on the expectation of the following expression
$$\mathbb{E}[\mathbf{X}^{-1}]-\alpha\mathbb{E}[\...
- 287
-1
votes
1
answer
138
views
On the concentration of Lipschitz functions near its expectation, where the vector has identical but not independent, components
Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent.
Now, I was wondering ...
- 1,467
4
votes
1
answer
214
views
Rates of convergence to Tracy-Widom?
$\renewcommand{\!}{\mathbf}
\renewcommand{\Ai}{\operatorname{Ai}}$
One can define the Tracy-Widom distribution as the Fredholm determinant $F_2(t)=\det(\mathbf I-\mathbf A)$ where
$$\mathbf A(x, y)=\...
- 63.9k
0
votes
0
answers
250
views
Concentration (or two sided tail bounds around expectations) of maximum and minimum of $n$ iid, subgaussian random variables
I asked this on MSE, but got no answer, hence asking here now. Help appreciated!
My question is motivated by this question and this question, where the first was aimed for giving a one sided tail ...
- 1,467
0
votes
1
answer
378
views
Concentration of norm of linearly transformed normal random vector as dimension go to infinity
Earlier asked on MSE, but didn't get an answer, so posting here:
Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $...
- 128k
0
votes
0
answers
141
views
What is the distribution of the norm of the multivariate $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d?$
Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||...
- 1,467
2
votes
1
answer
900
views
Asymptotically tight concentration of norms of subgaussian random vectors with independent coordinates, as the dimension $n \to \infty?$
Let $X=(X_1 \dots X_n)\in \mathbb{R}^n,$ be a subgaussian random vector so that $X_i$'s are independent, $\mathbb{E}X_i = 0, \mathbb{E}X_i^2=1.$ Before we pose our question, let's state the following:
...
- 1,467
0
votes
0
answers
132
views
Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix
Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
- 109
5
votes
1
answer
3k
views
Eigenvalues and eigenvectors of Gaussian random matrices
Let us assume we have a square matrix $A$ whose entries are sampled from a standard Gaussian distribution of mean $0$. Do we have any information about the distribution of its eigenvalues?
...
- 5,429
2
votes
1
answer
1k
views
Bound on eigenvalues of sample covariance matrices in terms of $d, n$, where $n=$ sample size, $d=$ dimension of data
Let $Z=[z_1, \dots z_n]$ be a $d \times n$ matrix, where the $z_i$'s are iid random vactors with mean $\mu \in \mathbb{R}^d$ and $d \times d$ (population) covariance matrix $\Sigma$, but the entries $...
- 188k
1
vote
0
answers
124
views
Law of large numbers and Central Limit Theorem for eigenvalues of perturbed matrices
I'm looking for results where perturbation by iid random entries to a matrix will result in convergence of the eigenvalues to the original eigenvalues. More precisely,
Let $ \forall n \in \mathbb{N},...
- 5,429
2
votes
1
answer
210
views
Marcenko-Pastur and Tracy-Widom laws for sample covariance and Gram matrices when the "features" are correlated: references
Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
- 188k
1
vote
1
answer
131
views
Large scale analysis of matrix multiplications
Let $\mathbf{A}_{m\times n}$ and $\mathbf{B}_{m\times n}$ be two random i.i.d matrices with zero mean and unit variance. Then, are the following large-scale analysis true (m,n go to infinity with ...
- 1,104
1
vote
0
answers
83
views
Tracy Widom type results for asymptotic distribution of the $k$-th largest eigenvalue of the sample covariance when $n, p \to \infty$?
Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here ...
- 1,467
2
votes
1
answer
760
views
Show the coordinate distribution has a very large sub-gaussian norm
Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e_i : i = 1,..., n\}$, where $e_i$ denotes the n-element set of the canonical basis vectors ...
- 23
2
votes
2
answers
854
views
Eigenvalue distribution of a random matrix
Is there any closed form distribution formula for the distribution of the eigenvalues of $\mathbf{X}^\mathrm{H}\mathbf{X}$ where the entries of $\mathbf{X}$ are independent Gaussian random variables ...
- 188k
0
votes
0
answers
274
views
Is there any relation between moments of random matrix and its eigenvalue distribution?
Let $\mathbf{X}$ be a random matrix with independent Gaussian random variable entries with different variances $v_{ij}$. Also define $\mathbf{A}=\mathbf{X}^\mathrm{H}\mathbf{X}$. Is there any relation ...
- 287
1
vote
1
answer
104
views
Limit of normalized sum of Dirac measures at first $\lfloor p/2\rfloor$ eigenvalues of the sample covariance matrix, with Marcenko-Pastur assumptions?
Let $\lfloor{*}\rfloor$ denotes the nearest integer $\le *$. I'm asking myself the question what's the limit of the part of the empirical spectral distribution corresponding to the first $\lfloor{p/2}...
- 188k
1
vote
1
answer
193
views
Random matrix properties
Let $\mathbf{H}_{N,K}$ be a random matrix whose entries are i.i.d complex Gaussian random variables with variance $1$. Then, we know from the law of large number that if $N,K\rightarrow\infty$, we ...
- 63.9k
0
votes
1
answer
320
views
Marcenko Pastur law when the dimensionality/sample size ratio $p/n \to 0, \infty$? Lack of resources?
Let $X: \Omega \to \mathbb{R}^{p \times n}$ be a random matrix so that each entry $X_{ij}$ is a random variable with $\mathbb{E}X_{ij}=0, \mathbb{E}X_{ij}^2=\sigma^2$
I was wondering what would ...
- 7,514
1
vote
1
answer
74
views
Joint density of a quadratic function of entries of orthogonal matrix
$U=(U_{ij})_{1\leq i,j\leq m},V=(V_{ij})_{1\leq i,j\leq m}$ are independently and uniformly distributed on the orthogonal group $O(m)$. For any positive integer $k,n$ such that $1\leq k\leq n\leq m$, ...
- 188k
7
votes
0
answers
179
views
Can one "smooth over" k-wise independence to get actual independence?
I came across the following toy problem and was curious if there was a simple solution or counterexample. Suppose you have a distribution $p$ on $m$ random variables $X_1, \ldots, X_m$, each with ...
2
votes
1
answer
206
views
Density of random matrix only depends on its spectrum
Suppose a random positive definite matrix $A\in\mathbb{R}^{n\times n}$ has density function (with respect to the lebesgue measure on $\mathbb{R}^{n(n+1)/2}$) $f(A)=g(\lambda_1(A),...,\lambda_n(A))$ ...
- 128k
0
votes
1
answer
681
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. $\mathcal{N}(0,1)$.
- 1,096