All Questions
16 questions
3
votes
1
answer
307
views
Request for references of random matrices
I need some good books aimed as a detailed and gentle introduction to random matrices, containing good discussion and derivation of Marchenko–Pastur distribution. Also, I request some other references ...
- 826
1
vote
1
answer
257
views
Using gradient descent in probability case
Suppose we have i.i.d. samples $x_i\sim N(0,\Sigma)$ and $y_i\sim x_i^T\omega^*+\xi_i,\xi_i\sim N(0,1)$ where $\omega^*$ is the fixed point of:
$$\omega_{i+1} = \omega_i − \eta\nabla_\omega f(\omega_i,...
- 421
3
votes
1
answer
373
views
Matrix positive semi-definite
We construct a non-random matrix using random variables as follows:
We fix the vector $v=(1,1).$
Let $X$ be a $\mathbb R^2$-valued random variable such that $X$ is distributed according to
$$d\mu(...
- 192
9
votes
1
answer
652
views
Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 124
3
votes
0
answers
184
views
Convergence rate of the smallest eigenvalue of an integral of a multivariate squared Brownian Motion
I am interested in deriving the convergence rate of the smallest eigenvalue of a sequence of random matrices with diverging dimension. More precisely, let $W_n(r)$ represent an $n$-dimensional ...
- 41
1
vote
1
answer
287
views
Random matrix and spherical spin-glass
The Hamiltonian of the p-spherical spin glass model is
$$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}$$
where $\sigma \in ...
- 536
2
votes
0
answers
123
views
Modified Wigner semicircle law
The Wigner semicircle law states that for a random GOE-matrix $M^N \in \mathbb{R}^{N \times N}$ in the $N \rightarrow \infty$ limit for any $f \in C^b(\mathbb{R})$
$$\lim_{N \rightarrow \infty}\frac{...
user118408
1
vote
0
answers
146
views
minimum eigenvalue of Katri-Rao product of two Gaussian matrices
Let $\mathbf{A}\in\mathbb{R}^{k\times n}$ and $\mathbf{B}\in\mathbb{R}^{d\times n}$ be independent matrices with i.i.d. $\mathcal{N}(0,1)$ entries. I'm interested in lower bounding the minimum ...
- 363
1
vote
0
answers
295
views
One-sided Talagrand concentration inequality for empirical processes
Let $\mathcal{F}$ denote a function class. A classic result by Talagrand states that
\begin{align*}
\mathbb{P}\bigg\{\sup_{f\in\mathcal{F}}\big|\sum_{i=1}^nf(X_i)-\mathbb{E}\big[\sum_{i=1}^nf(X_i)\...
- 859
4
votes
0
answers
416
views
concentration of functions of Gaussian processes
Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...
- 859
11
votes
8
answers
2k
views
Semicircle law universality elsewhere
Wigner's semicircle distribution is:
$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$
Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
- 4,952
2
votes
0
answers
1k
views
Random matrices whose limit gives exact Wigner surmise
Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write $\rho^W_0(s)=\...
- 4,952
1
vote
1
answer
257
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 ...
- 859
4
votes
1
answer
637
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 ...
- 4,952
2
votes
1
answer
2k
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 non-...
- 127
1
vote
0
answers
132
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 ...
- 11