All Questions
94 questions
19
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 ...
- 2,242
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 \...
- 12.9k
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 ...
CommunityBot
- 1
8
votes
2
answers
547
views
Concentration inequality for minimal eigenvalue of sample covariance
I was reading an article of matrix completion and met the following lemma
The concentration inequality for $\sigma_{\max}$ part is a standard result. However, I didn't find any results like the $\...
CommunityBot
- 1
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,...
CommunityBot
- 1
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 $\...
CommunityBot
- 1
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 ...
CommunityBot
- 1
22
votes
1
answer
1k
views
Random distance matrices
My question is motivated by the following recent paper:
Gadgil, Siddhartha; Krishnapur, Manjunath, Lipschitz correspondence between metric measure spaces and random distance matrices, Int. Math. Res. ...
- 6,367
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 ...
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
1
vote
1
answer
245
views
expectation of the function of Wishart matrix eigenvalues
For Given a $N×M$ random complex gaussian matrix $X$ where $M=XX^H$, let $\lambda_1>\lambda_2>\cdots>\lambda_N$ be the ordered eigenvalues of $M$ my objective is to get an estimation of
$$
f =...
1
vote
0
answers
80
views
Moments from characteristic function for matrices
When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
CommunityBot
- 1
3
votes
0
answers
131
views
Matrix-Gaussian distributions
The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
- 193
1
vote
1
answer
84
views
Limiting value of Stieltjes transform of sum of independent Wishart matrices
Let $n_1$, $n_2$, and $d$ positive integers tending to infinity such that $d/n_k \to \phi_k \in (0,\infty)$ and $n_1/(n_1+n_2) \to p \in (0,1)$. Let $X_k$ be an $n_k \times d$ random matrix with iid ...
- 6,853
0
votes
0
answers
42
views
Limiting value of trace of resolvent matrix involving two independent Wishart random matrices
Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
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$$
...
4
votes
2
answers
1k
views
Expectation of the trace of inverse of a Gaussian random matrix
Given a $N×M$ random complex gaussian matrix $X$ and $N×K$ random complex gaussian matrix $Y$ I'm interested in approximating the expectation expressed as:
\begin{align}
E[trace({(aX{X^H} + I)^{ - ...
- 6,853
1
vote
0
answers
72
views
Dimension-free sample complexity for the inverse of Gaussian sample covariance?
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in ...
1
vote
0
answers
57
views
Limiting value of expectation of trace of truncated Gram matrix
Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...
- 6,853
1
vote
0
answers
67
views
Random matrix theory: accounting for mean
Assume a random matrix, denoted as $X$, which is an $n$ by $T$ matrix, $T\geq n$. While I understand the typical scenario where the random variables $X_{ij}$ are sampled from a $\mathcal{N}(0,\sigma_{...
- 6,367
4
votes
1
answer
485
views
Expected norms of Wishart matrices
Suppose $x_i \stackrel{\text{i.i.d}}{\sim} \mathcal{N}(\mu,\Sigma)$. What can we say about dependence on $b$ of Frobenius/spectral norm quantities below?
$$f(b)=\left\|\frac{1}{b}\sum_{i=1}^b x_i x_i^...
CommunityBot
- 1
6
votes
1
answer
271
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=...
- 188k
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}...
- 128k
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 $...
- 2,242
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 ...
- 5,474
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
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
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 $\Sigma$:
$$...
- 2,242
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. ...
- 1,724
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 ...
- 188k
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 ...
- 188k
3
votes
1
answer
844
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
7
votes
1
answer
295
views
Closure of random rotations
Are matrix Fisher random variables closed under multiplication?
For those unfamiliar with the jargon, let me unpack the terms above and repose my question.
This is a question about probability ...
- 2,821
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
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.
...
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
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}}.$$
Is ...
- 128k
1
vote
1
answer
226
views
Orthogonal transformation of multivariate Bernoulli-Gaussian distribution
Actually, I have asked this question in https://math.stackexchange.com/questions/4330127/orthogonal-transformation-of-multivariate-bernoulli-gaussian-distribution, but I think mathoverflow might be ...
- 188k
1
vote
1
answer
104
views
Limiting value of $\dfrac{1_n^\top B^{-1} A B^{-1} 1_n}{d}$, where $A=WW^\top + a I_n$, $B = WW^\top + b I_n$, and $W \sim N(0,\Sigma_d)$
Let $n$ and $d$ be positive integers with
$$
n,d \to \infty, \quad n/d \to \rho \in (0,\infty).
$$
Let $\Sigma_d$ be a psd matrix such that
$\mbox{trace}(\Sigma_d) = 1$.
$\|\Sigma_d\|_{op} = \mathcal ...
- 1,724
2
votes
0
answers
51
views
Spectral approximation of $(XX^\top/d)\circ(X\Sigma_dX^\top/d)$ where $X$ is an $n \times d$ random matrix with iid rows from $N(0,\Sigma_d)$
Let $X \in \mathbb R^{n \times d}$ be a random matrix with iid rows from $N(0,\Sigma_d)$ where $\Sigma_d$ is a $d \times d$ psd matrix verifying w.h.p,
$\mbox{trace}(\Sigma_d/d)= 1$.
$\|\Sigma_d\|_{...
- 6,853
2
votes
1
answer
90
views
Asymptotics of $w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix
Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. ...
- 6,853
2
votes
1
answer
187
views
Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix
Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c ...
- 188k
4
votes
0
answers
75
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
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}
$...
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}...
CommunityBot
- 1
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