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Statistics of spectral properties of matrix-valued random variables.
6
votes
0
answers
198
views
Spectrum of $\prod_i^d \left(I-x_ix_i^T\right)$ for isotropic $x_i$
Suppose $x_i\in \mathbb{R}^d$ are IID isotropic random vectors with $\|x_i\|=1$ and matrix $A_d$ is defined as follows:
$$A_d=\prod_i^d \left(I-x_ix_i^T\right)$$
Is anything known about the spectrum …
1
vote
2
answers
297
views
Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$
Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5
$$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$
Exerci …
2
votes
1
answer
229
views
Expected norm of a product of Gaussian matrices
Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof?
$$E[\|C_n\|_F^2]=d^{n+1}$$
This fol …
20
votes
0
answers
3k
views
What does a product of many Gaussian matrices converge to?
Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$.
Is any …
0
votes
0
answers
93
views
Additivity of purity of random matrix products
Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as
$$\rho(M)=\frac{n \sum_i \sigma_i^4}{\l …
6
votes
1
answer
264
views
Spectrum asymptotics for a product of $k$ random matrices?
How does the spectrum of a product of $k$ random matrices behave around 0?
In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k …
0
votes
0
answers
60
views
Norms of Wigner matrices under power law decay
Suppose $\Sigma=\operatorname{diag}(h)$ where $h=(1^{-p},2^{-p},3^{-p},\ldots,d^{-p})$ and $p> 1$
$X$ is a matrix with $b$ rows sampled independently from $\operatorname{Normal}(0,\Sigma)$
Suppose $d$ …
6
votes
0
answers
279
views
Estimating $E[\operatorname{Tr}(ABABBA..)]$ for random shuffling of $A,B$?
How can I estimate the following value where $A,B$ are $d\times d$ matrices and expectation is taken over all random permutations of the product?
$$E_\text{shuffle}[\operatorname{Tr}\underbrace{AA\cdo …
1
vote
0
answers
58
views
Behavior of $\operatorname{Tr}[H(I-H)^s]$ for random positive definite $H$
Suppose $H$ is a random positive definite matrix normalized to have norm 1. Can we say anything about behavior of $f(s)$?
$$f(s)=\operatorname{Tr}[H(I-H)^s]$$
Taking $H=A^T A$ with entries of $A$ samp …
4
votes
1
answer
468
views
Expected norms of Wishart matrices
Suppose $x_i \stackrel{\text{i.i.d}}{\sim} \mathcal{N}(\mu,\Sigma)$. What can we say about dependence on $b$ of Frobenius/spectral norm quantities below?
$$f(b)=\left\|\frac{1}{b}\sum_{i=1}^b x_i x_i^ …
8
votes
0
answers
232
views
Decay of orthogonal contributions in a random set of vectors
Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\f …
1
vote
1
answer
274
views
Bound on $i$th largest eigenvalue in a large Erdos-Renyi graphs
Typical magnitude of $i$th largest eigenvalue of an Erdos-Renyi random graph seems to decay at least exponentially with $i$. Is there an analytic expression for the constant in the exponent, or a nice …
6
votes
0
answers
285
views
Dimension-free sample complexity for estimating Gaussian covariance
(also asked on math.se, with no answers)
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need sample covariance $\Sigma_m$ to be $\epsilon$-close to true covariance $\Sigma$:
$ …
2
votes
0
answers
205
views
When does an infinite product of random matrices have finite expected norm?
Given a finite set of matrices $A_i$, sample $n$ matrices uniformly with replacement and compute $f_n=\|A_1 A_2\cdots A_n\|^2$. When is the following limit finite?
$$\lim_{n\to \infty} E[f_n]$$
I'm es …