Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
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Complexity of calculating the expectation of $\operatorname{Tr} h(A)$, $A$ is a random matrix
$A$ is a $d_1\times d_1$ random matrix. Given $\{g_i\}~(1\leq i\leq n)$ iid Gaussian variables, $f_{ij}(g_1,g_2,...,g_n)~(1\leq i,j\leq d_1)$ are degree-$d_2$ polynomials. And $f_{ij}\equiv f_{ji}~(\...
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Comparing the minimum squared Mahalanobis norm of $n$ [orthonormal vs. i.i.d] random unit vectors
Let $Q$ diagonal with coefficients $q_1 \geq ... \geq q_n \geq 0$.
Let $U = \left[ u_1 ~ ... ~ u_n \right]$ distributed uniformly on the set of $n \times n$ orthonormal matrices, and $\tilde{u}_1, ...,...
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Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression
From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...
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Trace of product of two Wishart matrices
Let $A,B$ be two independent complex Wishart matrices, $A,B\sim CW_p(\mathbf{I},n)$, that is $A=\frac1n GG^\dagger$& $B=\frac1n QQ^\dagger$ where $G$ and $Q$ are independent $p\times n$ complex ...
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Random walk on matrix until singularity
Consider a random walk on matrices, where one starts with the matrix $M=I_n$ and at each step randomly chooses an entry of $M$ to increase by $1$.
I’m interested in two things about this walk:
What’s ...
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Eigenvalues of the 1D lattice with diagonal disorder?
Consider the adjacency matrix $\mathbf{A}$ of a one dimensional lattice of size $N$. That is, $A$ is a $N\times N$ matrix with $A_{ij}=1$ if vertex $i$ adjacent to vertex $j$ (there exists an edge ...
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Uniform distribution on pairs of unitary matrices
This question has two parts.
In Part 1, I would like to know if the following distribution on pairs of $d$-dimensional unitary matrices has popped up in the literature:
"Uniform distribution on ...
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Why are singular values of random matrix $[X \mid Y] \in \mathbb{R}^{N\times 2T}$ so close to those of $XY^T \in \mathbb{R}^{N \times N}$, $X\sim Y$
As an accidental byproduct of some numerical simulations I have been doing as part of a research paper in machine learning, I made the observation that the singular values of the random matrix $\frac{...
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Smallest Singular Value of submatrices of a column-orthogonal matrix
Suppose we have a column-orthogonal matrix $\mathbf {U}\in\mathbb{R}^{n\times p}$, satisfying $\mathbf {U}^{\top}\mathbf {U}=\mathbf {I}_p$. We select $m<n$ rows of $\mathbf {U}$ randomly and get $\...
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Random matrices may be asymptotically free but never free themselves?
It is well known that independent $N\times N$ unitarily-invariant random matrices (or independent families of random matrices) may be asymptotically free as $N\to \infty$ with respect to the ...
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Lower tail of random rank one sums?
Let $\{x_i\}_{i\geq 1}$ be iid random elements of the sequence space $\ell^2(\mathbb{N})$;
assume that $\|x_i\|_2 \leq 1$ almost surely. Let $\Sigma = \mathbb{E}[x_1 \otimes x_1]$.
Define
$$
\Sigma_n =...
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Is the exact formula for the trace moments of an isotropic complex Wishart matrix known?
Let $\mathbb{X}_{p,n}$ be a $(p \times n)$ random matrix whose entries are iid complex standard normal random variables. The hermitian random matrix $\mathbb{S}_{p,n} = \frac{1}{n} \mathbb{X}_{p,n} \...
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Expectation of the trace of random matrix with an inverse insided
Consider a $N$-dimensional random complex vector $\mathbf{x} \in \mathbb C^{N \times 1}$ following the complex Gaussian distribution, i.e., $\mathbf{x} \sim {CN}(0,\sigma^2 \mathbf{I})$, where $\...
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What does a product of many Gaussian matrices converge to?
Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$.
Is ...
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Singular values of Kronecker product of random matrices
I'm looking for a way to evaluate $\mathbb{E} \| (\mathbf{X} \mathbf{Q})^+ \|$ for a random matrix $\mathbf{X} \in \mathbb{R}^{r \times n}$ and a (fixed) matrix $\mathbf{Q} \in \mathbb{R}^{n \times \...
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Expected value of MGIG distribution
I'm currently dealing with a Gibbs sampler of the multivariate generalized inverse Gaussian distribution (MGIG). In order to check the correctness of the sampler, I'd like to know the expected value ...
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Probability that two random gaussian matrix will have large distance
I have two independent random gaussian matrix $A$ and $B \in R^{d\times n}$, and i want to compute an upper bound of the probability that
$$Pr(\left\| (A-B)(A+B)^T\right\| \leq a)$$
One method might ...
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Distribution of covariance matrix of vectors of lognormal rvs
I know that the Wishart distribution is the distribution of the sample covariance matrix of a multivariate normal distribution. I am wondering if the analog distribution for a sample from a ...
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Random matrix models with squared singular value density diverging as $x^{-2}$ at 0
Are there named RMT models where the density of squared singular values diverges as $x^{-2+\epsilon}$ around 0?
Ginibre matrix has $x^{-\frac{1}{2}}$ decay, an infinite product of such brings it to $x^...
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Expected value of the projective metric between random orthogonal Stiefel matrices in $\mathbb{R}^{N \times k}$ equals $1 - \frac{k}{N}$
This is a cross-post from this other question that I asked ~1 month ago in the mathematics forum, with no reaction. I am still stuck on this, looking for references or approaches to proofs. I hope I ...
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Anti-concentration inequality for the eigenvalue of Gaussian matrix
Let $f(x) = f(x_1, . . . , x_n)$ be a polynomial of degree $d$ and $\text{Var}[f] = 1$. One result by Carbery and Wright shows that for any $t\in\mathbb{R}$ and $ε > 0$,
$$
\text{Pr}_{x\sim N^n}[|f(...
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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}{\...
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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=...
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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 ...
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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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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.
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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 ...
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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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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
$$\...
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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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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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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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