All Questions
9 questions
6
votes
3
answers
1k
views
Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements
Consider a random matrix $A \in \mathbb{R}^{N \times N}$ where the elements are random gaussian variables. The mean and variance of the elements are different on the diagonal and the off-diagonal:
$\...
- 355
5
votes
1
answer
1k
views
A general formula for Gaussian integrals over matrix elements
The question I have is quite specific. So in the hope that this post might help others in the future, my problem boils down to solving the following integral:
$$I_\tau=\int \prod_{i, j=1}^{N} d J_{i ...
- 117
3
votes
1
answer
146
views
Orthogonal projection $X X^+$ from random Gaussian matrix $X$
Given a standard Gaussian matrix $X\in\mathbb{R}^{n\times d}$, $d<n$, with entries sampled i.i.d. from $\mathcal{N}(0,1)$, is the corresponding orthogonal projection $X X^+ = X (X^\top X)^{-1} X^\...
3
votes
0
answers
131
views
Matrix-Gaussian distributions
The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
- 193
2
votes
3
answers
999
views
Sum of Square of the Eigenvalues of Wishart Matrix
Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
- 1,096
1
vote
1
answer
287
views
Random matrix and spherical spin-glass
The Hamiltonian of the p-spherical spin glass model is
$$H_{N,p}(\sigma)=\frac{1}{N^{\frac{p-1}{2}}} \sum_{i_1,...,i_p=1}^N X_{i_1,...,i_p} \sigma_{i_1}\cdot...\cdot \sigma_{i_p}$$
where $\sigma \in ...
- 536
0
votes
2
answers
135
views
Expectation of supremum of sub gaussians
I am trying to prove Lemma 2.3 of ON THE SPECTRAL NORM OF
GAUSSIAN RANDOM MATRICES, which states that
Let $X_1,\cdots,X_n$ be not necessarily independent random variables with $\mathbb{P}[X_i > x] ...
- 150
0
votes
1
answer
73
views
Algorithm for economically sampling method for Gaussian matrix product
Let $A$ be an $n\times n$ random matrix with i.i.d. $N(0,\sigma)$ entries, for some $\sigma>0$ and let $x\in \mathbb{R}^n$. A direct computation shows that $Ax \sim N(0,\sigma x^{\top}x)$.
I would ...
- 5,405
0
votes
0
answers
321
views
Projecting a vector onto a random subspace
Let $A\in\mathbb{R}^{k\times d}$ be matrix with i.i.d. $\mathcal{N}(0,1/k)$ entries with $k<d$, and let $B=A^{\top}A$. I would like to compute the distribution of $Bx$ where $x\in\mathbb{R}^{d}$ is ...
- 129