Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
841
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Expectation of the operator norm of projection of a random permutation matrix
Assuming I have a fixed dimension $p$ subspace of $\mathbb{R}^d$ orthogonal to $1^d$ and $VV^\top$ with $V \in \mathbb{R}^{d \times p}$ is the orthogonal projection to the subspace.
What bound can I ...
2
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2
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How to analyze the value of convergence of functions of random matrices?
Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of ...
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63
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Spectrum of Moore-Penrose pseudo-inverse multiplied by a constant
Consider a random rectangular matrix $X\in\mathbb{R}^{N\times P}$ where each entry is drawn from iid distribution with mean $m$ and variance $s^2$, and denote $X^+$ the Moore-Penrose pseudo-inverse.
...
3
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1
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Taylor expansion of Stieltjes Transform
I'm trying to derive a very basic result stated in several books on random matrix theory (e.g. Terry Tao's book and Potters & Bouchaud's book).
Given a symmetric matrix $A \in \mathbb{R}^{N \times ...
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A particular selection of rows in upper triangular matrices
Let $A$ be a strictly upper triangular $n\times n$ matrix whose entries are either 0 or 1 (diagonal entries are all 0) with the nullity $m<n$.
Let us denote $R_j$ and $C_j$ with the rows and ...
3
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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 ...
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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 ...
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Asymptotics of a certain trace involving random matrices with general elliptical covariance structure
Let $n,d,m$ be large positive integers that the ratios $d/n$ and $d/m$ are fixed in $(0,\infty)$. Let $G \in \mathbb R^{n \times d}$ and $S \in \mathbb R^{d \times m}$ be independent random matrices ...
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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 ...
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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}...
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171
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matrix bernstein's inequality: from tail probability to expectation
Let $X_i$ be independent, mean zero, $n\times n$, symmetric random matrices. $\|X_i\|\leq K$ almost sure for $\forall I$.
We have matrix Bernstein's inequality for the tail probability as follows
$$\...
2
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Trouble understanding a Lemma in Pastur's Paper
I'm having trouble understand Eq 3.51 Lemma 3.3 in https://arxiv.org/pdf/2001.06188.pdf
The basic premise is
$$\begin{align}
&\eta _{j}(t)=t^{1/2}\eta _{j}+(1-t)^{1/2}q_{n}^{1/2}\gamma _{j},
\;t \...
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Existence of a matrix with bounded entries and large smallest singular value
Is the following statement true?
For all $n$ large enough, there exists an $M_n \in [-1,1]^{n \times n}$ such that the smallest singular value of $M_n$, $\sigma_n(M_n) \gtrsim \sqrt{n}$.
If $n$ is ...
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Spectral universality of sample covariance from unit sphere
If $x_1,\dots,x_n\sim \mu$ are mean-zero iid samples in $R^d$ that are drawn from unit sphere, with covariance $E x x^\top = I_d,$ and $C_n:=\frac1n \sum_i^n x_i x_i^\top$ is the sample covariance, ...
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The probability that the dominant eigenvalue of a random real matrix is real
Let $X_n$ be an $n\times n$ real matrix where the entries in $X_n$ are independent, normally distributed, have mean $0$, and variance $1$. Suppose that $\lambda_1,\dots,\lambda_n$ are the eigenvalues ...
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What is the distribution of the matrix elements for a Poisson distribution of eigenvalue spacing?
I've already know that entries with normal distribution will give the Wigner-Dyson distribution of eigenvalue spacing, but what about the Poisson distribution of level spacing? What kind of random ...
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The limit spectral distribution of the random matrix $(\hat{\Sigma}_1+\hat{\Sigma}_2)^{-1}\hat{\Sigma}_1$
Let $S_1$ and $S_2$ be the collection of i.i.d. copies of $X\sim\mathcal{N}(0,I_p)$, where $|S_1|=n_1,|S_2|=n_2$. Let $\hat{\Sigma}_1$ and $\hat{\Sigma}_2$ be the covariance matrix using samples in $...
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Sum of entries of $W^k$ in terms of limiting spectral density of $W$?
Suppose $h$ is spectrum of a random matrix $M$ and $e$ is a vector valued time-series in $\mathbb{R}^d$ with $d\approx \infty$, which starts with $(1,1,\ldots,1)$ and updates $i$'th component at each ...
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Large deviation principle for product of iid bounded symmetric random variables
Let $n$ and $k$ be positive integers. Let $X$ be the empirical mean of $n$ iid Rademacher random variables. Note that the distribution of $X$ is symmetric about 0, and also $|X| \le 1$ w.p 1. Let $X_1,...
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Concentration of a certain simple / well-structured random multilinear polynomial with growing degree
Let $k$ and $N_1$ be positive integers and set $N=kN_1$. Partition $[N] := \{1,2,\ldots,N\}$ $k$ disjoint from $G_1,\ldots,G_k$ of each of size $N_1$, and let $\mathcal T(k,N_1)$ be a transversal of ...
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Maximum norm within a random subspace intersected with an ellipsoid
Let $d < n$, and let $G_n(d)$ denote the space of all $d$-dimensional subspaces of $\mathbb{R}^n$.
Let $a = (a_1,\dots, a_n)$ denote a positive sequence, and define
$U(a) = \{u \in \mathbb{R}^n: \...
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1
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Density of eigenvalues of empirical covariance matrix of vectors uniform on the sphere
Is anyone able to point me to a reference for this?
Let the rows of $X \in \Re^{n\times d}$ be i.i.d. uniform on the sphere of radius $\sqrt{d}$ in $\Re^d$. What is the density of the eigenvalues of $...
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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 $...
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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?
$$E_\text{shuffle}[\operatorname{Tr}\underbrace{AA\...
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Concentration of a combinatorial sum
Let $X=(x_1,\ldots,x_p)$ be an $p \times n$ random matrix with iid entries from $\{\pm 1\}$, distributed so that $\mathbb P(x_{ij} = 1) \equiv 1/2$, where $x_i=(x_{i1},\ldots,x_{in})$. Let $y$ be a ...
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Behavior of $\operatorname{Tr}[H(I-H)^s]$ for random positive definite $H$
Suppose $H$ is a random positive definite matrix normalized to have norm 1. Can we say anything about behavior of $f(s)$?
$$f(s)=\operatorname{Tr}[H(I-H)^s]$$
Taking $H=A^T A$ with entries of $A$ ...
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1
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How to bound $ P(\frac{1}{N}\sum_{i=1}^N \sigma^i X_i^2\ge ax)$ for eigenvalues of a normalized $N\times N$ GOE matrix?
Let $A$ be a normalized $N\times N$ GOE matrix. Let $\sigma^1<\sigma^2<\dots <\sigma^N$. We know that the largest eigenvalue converges $\sigma^N$ to 2 almost surely. Assume that $X_1,\dots, ...
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1
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111
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The probability upper bound on the ratio of the eigenvalues
Consider a $N\times N$ normalized matrix sample from GOE (the definition see https://www.lpthe.jussieu.fr/~leticia/TEACHING/Master2019/GOE-cuentas.pdf). If we apply the following result of the edge of ...
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1
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Calculating $\mathbb{E}\left(\tfrac{XX^T}{\|AX\|^2}\right)$ for isotropic random vectors $X$
$\newcommand{\EE}{\mathbb{E}}$
Let $A\in\mathbb{R}^{m\times n}$ and $X$ be an isotropic random vector in $\mathbb{R}^n$, i.e. it holds that $\EE(XX^T) = I_n$.
How to calculate $$M = \EE\left(\tfrac{XX^...
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Can we show that for every $\delta>0$, there exist constants $\alpha>0, \beta>0$ so that the following inequality holds with high probability?
Consider two $n-$dimensional random vectors $u$ and $v$ uniformly distributed on the sphere. Define $X_n :=u\cdot v$. Note that as $n\to \infty$, $\sqrt{n}X_n \to N(0,1)$ as $n\to \infty$. Fix $\...
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1
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Does there exist a constant $c>0$ such that $$ P(N^{2/3}(\lambda_N-\lambda_{N-1})\ge c)\ge 1-\epsilon? $$
Following this question: Can we get that $ P(N^{2/3}(\lambda_N-\lambda_{N-1})\le c)\ge 1-\epsilon$?.
We know that for $\lambda_N\le \lambda_{N_1}\le \dots le\lambda_1$ (eigenvalues of GOE matrix)
$$
\...
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CLT of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional random matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\...
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1
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Bound for an expectation of random matrix with quantized random variable
Let random matrix $\mathbf{X} \in \mathbb{C^{\mathrm{m} \times \mathrm{n}}}$ and random vector $\mathbf{y} \in \mathbb{C^{\mathrm{m} \times 1}}$ are unknown distributed, but their covariance and ...
1
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1
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131
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Bound for expectation of random matrix
Let random matrix $\mathbf{X} \in \mathbb{C^{\mathrm{m} \times \mathrm{n}}}$ and random vector $\mathbf{y} \in \mathbb{C^{\mathrm{m} \times 1}}$ are unknown distributed, but their covariance and ...
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Tail bound on largest singular value of Gaussian Wigner matrix
I have problem on deducing the following tail bound on largest singular value of Gaussian Wigner matrix
$\|W\|\leq(2+\epsilon)\sqrt{n}$, $\forall\epsilon$, with high probability.
There is a hint: see ...
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1
answer
75
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Asymptotic property of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top.
...
2
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1
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Generating a random matrix with large spark (i.e., each $k$-tuple of columns is linearly independent)
Let $F$ be a field, and let $m, n, k$ be positive integers. Is there an efficient algorithm to compute a uniformly random $m \times n$ matrix $A$ over $k$ such that each $k$-tuple of columns of $A$ is ...
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1
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71
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Control the summation of a diagonal matrix and another matrix to be full rank
Statement. To ensure the rank of $\operatorname{ddiag}(AQQ^T)-\sigma\Delta=n$, it is sufficient to require $\min_i(\operatorname{diag}(AQQ^T))_i>\sigma\lVert\Delta\rVert$.
Note: $Q\in\mathbb{R}_{n\...
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1
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Is svd of a Gaussian iid matrix corresponds to Haar measure on the Stiefel manifold?
I want to draw a matrix $A\in \mathbb{R}^{n\times k}$ uniformally at random from the Stiefel manifold $\mathbb{V}_k(\mathbb{R}^n)$, that is from the collection of all $n\times k$ matrices $A$ such ...
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Control the largest eigenvalue of random matrix
Objective: If there exists $\epsilon > 0$ such that $\sigma <
\sqrt{\frac{n}{(2+\epsilon)\log n}}$
then, with high probability, we have $\lambda_{\text{max}}(D_{[-W]}+W)<\frac{n}{\sigma}$. $W$...
2
votes
1
answer
91
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Deviation of random matrix from its expectation informs the positiveness of its second smallest eigenvalue
Let $A$ be a PSD random matrix, which has $0$ as one of its eigenvalues. The second smallest eigenvalue of the expectation of $A$ writes as $\lambda_2(\mathbb{E}(A))>0$.
Why the following statement ...
4
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1
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Joint distribution of minor of Wigner Hermitian matrices
Let $A$ be an $n\times n$ random matrix with i.i.d entries (say standard Gaussian) $A_{ij}$. I know that there is a CLT type result known for the determinant of $A$. More precisely there is a CLT for $...
1
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1
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75
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Is this Markov chain of Gaussian matrix products $G_1 G_2 \dots G_m$ ergodic?
Consider Markov chain $\{X_t\}_{t\in N}\subseteq R^{n\times n}$ defined by $X_{t} = X_0 G_1 \dots G_t$ where $G_i$'s are iid Gaussian matrices $G_1,\dots,G_t\sim N(0,1/n)^{n\times n}$, and $X_0$ is ...
1
vote
1
answer
164
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Distribution of inner product of random unit vectors
Suppose I have two $2\times2$ Haar random unitary matrices $u_1$ and $u_2$, then I can define a diagonal matrix $$\begin{pmatrix}(u_1\cdot u_2)_{11}&0\\ 0&(u_1\cdot X\cdot u_2)_{11}\end{...
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0
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93
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Induced distribution from partial trace of random unitary
I am sure this must have been covered in the mathematical literature (and is certainly related to a similar question I had asked previously), but hoping someone can direct me to the right place.
Let ...
0
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1
answer
111
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Can we still have the order of ratio result of the two smallest eigenvalues?
For GOE matrix $A$, we have the following limiting distribution for eigenvalues of $A$ by $\lambda_N\ge \lambda_{N-1}\ge \dots \ge \lambda_1$:
In this [paper][1], if we denote the $k$ largest ...
2
votes
1
answer
737
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Can we get that $ P(N^{2/3}(\lambda_N-\lambda_{N-1})\le c)\ge 1-\epsilon$?
Following this question: Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy-Widom law?.
We know that
$$
\lim_{N\to\infty}P(N^{2/3}(\lambda_N-2)\le s_1,\dotsc,...
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1
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Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy-Widom law?
In this paper, if we denote the $k$ largest eigenvalues by $\lambda_N,\lambda_{n-1},··· ,\lambda_{N-k+1}, $ then for Gaussian ensembles the joint distribution function of rescaled eigenvalues has the ...
4
votes
1
answer
610
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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 $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
2
votes
2
answers
173
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Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions
Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent).
Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt ...