All Questions
94 questions
1
vote
0
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80
views
Moments from characteristic function for matrices
When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
- 193
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
1
vote
1
answer
84
views
Limiting value of Stieltjes transform of sum of independent Wishart matrices
Let $n_1$, $n_2$, and $d$ positive integers tending to infinity such that $d/n_k \to \phi_k \in (0,\infty)$ and $n_1/(n_1+n_2) \to p \in (0,1)$. Let $X_k$ be an $n_k \times d$ random matrix with iid ...
- 6,853
0
votes
1
answer
108
views
RMT for modified Wishard matrix $Y'Y$ (where $i$th row of $Y$ is zero if $|x_i^\top u| \le \theta$; else it equals $x_i$)
Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\...
- 6,853
0
votes
0
answers
42
views
Limiting value of trace of resolvent matrix involving two independent Wishart random matrices
Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
d/n_k \to \phi_k \in (0,\infty).
$$
Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
- 6,853
1
vote
2
answers
306
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
$$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$
...
- 2,252
1
vote
0
answers
72
views
Dimension-free sample complexity for the inverse of Gaussian sample covariance?
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in ...
- 517
1
vote
0
answers
57
views
Limiting value of expectation of trace of truncated Gram matrix
Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...
- 6,853
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_{...
20
votes
0
answers
3k
views
+200
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 ...
- 2,252
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 ...
- 2,560
6
votes
1
answer
274
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=...
- 2,252
2
votes
0
answers
96
views
Limiting value of $\dfrac{1}{m}\mathrm{tr}(FAF^\top (FBF^\top)^{-1})$, where $F$ has iide $N(0,1)$ entries and $A,B$ are deterministic
Let $F=F_{m,d}$ be a random $m \times d$ matrix with iid entries from $N(0,1)$. Let $A=A_d$ and $B=B_d$ be deterministic $d \times d$ positive-definite matrices. In case it helps, it may be assumed ...
- 6,853
1
vote
0
answers
68
views
Limiting value of expectation of $\operatorname{tr}(BR(z))$, where $R(z) := (X^\top X - z I_d)^{-1}$ and $X \sim N_{n,d}(0,A)$
Let $A=A(d)$, and $B=B(d)$ be (sequences of) deterministic positive-definite $d \times d$ matrices and let $X$ be an $n \times d$ random matrix with iid rows from $N(0,A)$. Let $R$ be the resolvent of ...
- 6,853
0
votes
1
answer
77
views
Estimation on rotationally-disturbed random vectors
During developing a new statistical estimator, I faced the following problem.
Let $\mathbf{x}_i$ be a sequence of i.i.d. $d$-dimensional random vectors with
\begin{align*}
\mathbf{x}_i = \mathbf{O}...
- 284
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 $...
- 2,252
4
votes
1
answer
626
views
How to get the lower bound of the following $\tau$?
Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
- 288
2
votes
2
answers
228
views
Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$
Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly ...
- 6,853
3
votes
1
answer
206
views
Asymptotic results for smallest gap of Gaussian random matrix
For a symmetric Gaussian random matrix $G=\{G\}_{1\le i,j \le n}$ with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=1/n$ (it is normalized), ordering its eigenvalues $\lambda_1\le \lambda_2\le\cdots \lambda_n$.
...
- 288
4
votes
1
answer
485
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^...
- 2,252
0
votes
1
answer
108
views
On the invertibility of $Z^\top Z$, where $Z$ is a Random matrix with concentrated weakly correlated entries
Let $d$, $n$, and $m$ be large positive integers. Let $X=(x_1,\ldots,x_n) \in \mathbb R^{n \times d}$ be a random matrix iid rows from some distribition $P$ on $\mathbb R^d$ which admits a density. ...
- 6,853
2
votes
1
answer
936
views
Upper-bound for spectral norm of the covariance matrix of a certain Gaussian vector with correlated entries
Let $n$ and $m$ be large positive integers. Let $x=(x_1,\ldots,x_n)$ be a vector of independent random variables from $N(0,1)$. It is clear that the covariance matrix of $x$ is $I_n$, the identity ...
- 6,853
3
votes
1
answer
111
views
Distribution/moments of transformed normally distributed random vector
Let $\varepsilon \sim N\left ( 0,I_{k} \right )$, consider the following function of $\varepsilon$,
$y=\left ( A+B\varepsilon \varepsilon {}'B{}' \right )^{^{\frac{1}{2}}}\varepsilon $,
where $A$ is a ...
- 31
4
votes
1
answer
164
views
Limiting value of expectation of trace of exponential of Wishart matrix
Let $X$ be an $n \times d$ random matrix with iid entries from $N(0, 1/d)$. Let $S:=X^\top X/n$, a $d \times d$ Wishart matrix and let $T = e^{S} := \sum_{k=0}^\infty \dfrac{S^k}{k!}$ be its ...
- 6,853
3
votes
0
answers
93
views
Explaning why the spectrum of a setting simple structure random matrix is always spiked ($d-1$ eigenvalues close to zero, and $1$ away from zero)
For concreteness, let $m=500$, $d=600$, $N=1000$. Let $W$ be and $d \times m$ matrix with unit-norm rows and let $u$ be a uni-norm vector of length $m$. Given a binary vector $b$ of length $m$, length ...
- 6,853
0
votes
0
answers
91
views
Spectrally-weighted Stieltjes transform of random matrix $Z=XX^\top$ in terms of Stieltjes transform of $Z$ and the weighting function
Let $n$ and $d$ positive integers going to infinity such that $d/n \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ iid rows from $N(0,\Sigma)$, where $\Sigma = diag(\lambda_1,\ldots,\...
- 6,853
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 ...
- 21
1
vote
1
answer
104
views
Limiting value of $\dfrac{1_n^\top B^{-1} A B^{-1} 1_n}{d}$, where $A=WW^\top + a I_n$, $B = WW^\top + b I_n$, and $W \sim N(0,\Sigma_d)$
Let $n$ and $d$ be positive integers with
$$
n,d \to \infty, \quad n/d \to \rho \in (0,\infty).
$$
Let $\Sigma_d$ be a psd matrix such that
$\mbox{trace}(\Sigma_d) = 1$.
$\|\Sigma_d\|_{op} = \mathcal ...
- 6,853
1
vote
1
answer
157
views
Moments of rescaled Bernoulli random matrix
Suppose $X \in \{0,1\}^{n \times m}$ is a matrix generated according to the following generative process:
$$Z_{ij} \sim \text{Bernoulli}(p) \implies X_{ij} = \frac{Z_{ij}}{\sum_{k=1}^m Z_{ik}}.$$
Is ...
- 269
2
votes
0
answers
51
views
Spectral approximation of $(XX^\top/d)\circ(X\Sigma_dX^\top/d)$ where $X$ is an $n \times d$ random matrix with iid rows from $N(0,\Sigma_d)$
Let $X \in \mathbb R^{n \times d}$ be a random matrix with iid rows from $N(0,\Sigma_d)$ where $\Sigma_d$ is a $d \times d$ psd matrix verifying w.h.p,
$\mbox{trace}(\Sigma_d/d)= 1$.
$\|\Sigma_d\|_{...
- 6,853
1
vote
1
answer
160
views
Given iid $w_1,\dotsc,w_N \sim N(0,1/d)$ iid, find a simple matrix $A$ s.t $\|aa^T-A\|_\text{op}\to0$, where $a_i := E_{G \sim N(0,1)}[f(\|w_i\| G)]$
Let $d$ and $N$ be two large comparable integers, for example assume
$$
N,d \to \infty, \quad d/N \to \gamma \in (0,\infty).
$$
Let $w_1,\dotsc,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \...
- 6,853
1
vote
1
answer
415
views
Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)
Let $g:\mathbb R \to \mathbb R $ be a continuous function which is
"sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and
"sufficiently integrable" (e.g integrable w.r.t $N(0,...
- 6,853
2
votes
1
answer
90
views
Asymptotics of $w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix
Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. ...
- 6,853
2
votes
1
answer
187
views
Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix
Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c ...
- 6,853
4
votes
1
answer
332
views
Asymptotic limit of trace of random matrix $(aI_m + WW^\top)^{-1}$, where $W$ has iid rows from $N(0,\Sigma)$
Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a ...
- 6,853
4
votes
0
answers
75
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
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}
$...
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\...
- 219
1
vote
1
answer
126
views
Probabilistic lower and upper-bounds for a certain random quartic form involving gaussian random matrices
Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C$ be a $d \times d$ psd matrix with $trace(C)=\mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the ...
- 6,853
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
$$\mathbf{\Gamma}_{n\times k}...
user274127
6
votes
0
answers
295
views
Dimension-free sample complexity for estimating Gaussian covariance
(also asked on math.se, with no answers)
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need sample covariance $\Sigma_m$ to be $\epsilon$-close to true covariance $\Sigma$:
$$...
- 2,252
3
votes
1
answer
845
views
Concentration inequality for the sample covariance matrix
I'd like to know if there is a concentration inequality for the sample covariance matrix that don't assume the knowledge of the true mean.
Background.
Given a probability distribution $\mu$ on $\...
- 903
2
votes
0
answers
172
views
Asymptotic lower and upper bounds for the eigenvalues of hadamard product $W \circ W$, where $W$ is a large Wishart matrix
Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic ...
- 6,853
3
votes
2
answers
885
views
Expectation of product of random matrices
Let $X$ and $Y$ be independent random symmetric matrices. What can one say about $\mathbb{E} [X Y X Y]$ or $\mathrm{trace} \mathbb{E} [X Y X Y]$ in terms of properties of $X$ and $Y$?
In particular, ...
- 33
2
votes
1
answer
415
views
High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)
Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
- 6,853
1
vote
1
answer
144
views
Bounds for the extreme singular-values of random matrix with thresholded entries
Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random ...
- 6,853
2
votes
1
answer
668
views
Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$
Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w_1,\ldots,w_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I_d)$, and define the $k ...
- 6,853
3
votes
1
answer
379
views
Concentration inequality for norm of solution to nonlinear least-squares problem
Define the piecewise-linear function $\psi(t):=\max(t,0)$ for all $t \in \mathbb R$.
Let $d,n,k \to \infty$ at the same rate (i.e $n \asymp k \asymp d$).
Let $y_1,\ldots,y_n \in \{-1,1\}$ uniformly ...
- 6,853
3
votes
1
answer
902
views
Expectation of exponential of Gaussian random matrix
Let $X$ be an $(N, M)$ random Gaussian matrix where $M<N$. For a given vector $v$, I want to estimate the expectation of:
\begin{align}
E\left[ {{v^T}X{X^T}{v}} \right]
\end{align}
and
\begin{align}...
- 377
3
votes
0
answers
230
views
Expectation of angle between two vectors in the image of a gaussian random matrix
Let $m$ and $n$ be large positive integers (going to infinity), and let $W$ be a random matrix of size $n \times m$ with iid entries from $N(0,1/m)$. Let $x,y \in \mathbb R^m$ be deterministic vectors,...
- 6,853
2
votes
2
answers
303
views
Expectation of the determinant of the inverse of non-central Wishart matrix
Let $A$ be $(n,n)$ central Wishart matrix with $k$ degrees of freedom.
my question is there is a way to estimate the expectation of:
\begin{align}
E[det(I+(I+A)^{-1})]
\end{align}
- 377