Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
307 questions with no upvoted or accepted answers
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Independence of random projection and orthogonal projection
Suppose we have three fixed unit vectors $x, y, z \in \mathbb{R}^d$ and an (arbitrary) distribution over random matrices $M \in \mathbb{R}^{k \times d}$: let $P_M = M^T(MM^T)^{-1}M$ and $P^{\perp}_M = ...
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Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
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Concentration for $\sum_{i=1}^n y_i \psi(x_i^\top u)$, for $y_1,\ldots,y_n \sim \{\pm 1\}$ and $x_1,\ldots,x_n$ uniform iid on hypersphere
Let $y_1,\ldots,y_n$ be drawn iid uniformly from $\{\pm 1\}$ and let $x_1,\ldots,x_n$ be drawn iid uniformly from the unit-sphere $(d-1)$-dimensional sphere $\mathbb S_{d-1}$, and independently from ...
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About symmetric rank-1 random matrices
Consider a $2n-$dimensional symmetric random matrix $M$ of form, $M = \begin{bmatrix} aa^T & ab^T \\ ba^T & bb^T \end{bmatrix}$ where $a$ and $b$ are $n$ dimensional random vectors.
Are there ...
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Good lower-bound for $\inf_{x \in \Delta_n} \|Gx\|$ where $G$ is an $N \times n$ random matrix with iid entries from $\mathcal N(0,1/\sqrt{N})$
Let $G$ be an $N \times n$ random matrix with independent entries distributed according to a centered Gaussian with variance $1/\sqrt{N}$ and let $n/N = \lambda \in (0, 1)$. Let $\Delta_n$ be the $(n-...
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Spectral properties of finite random matrices with regards to Kurtosis
Given a $N \times M$, $N\ge M$ finite random matrix where the elements are drawn from a probability distribution with Kurtosis $\gamma$.
Is there anything that can be said about the singular values (...
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Random matrix invertible
I am trying to figure out why the following random matrix is invertible:
\begin{align*}
A_j = I_d + J_{\mu}(\hat{X}_{t_{j-1}})(t_j - t_{j-1}) +
\left( \begin{array}{rrr}
B_1^T \\ ...
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126
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A packing ball problem: verify lower bound on Gaussian width of sparse ball
Note: This should be a geometry problem about packing balls. All the necessary probability pre-requisite is given below.
Consider a set of sparse vectors: $T_{n,s}:=\{x\in \mathbb{R}^n:\|x\|_0 \le s, \...
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Is this inequality of the random matrices correct?
I need to confirm whether the following inequality correct.
Let $\xi_i\in\{\pm 1\}$ be independent random signs, and let
$A_1,\ldots, A_n$ be $m\times m$ Hermitian matrices. Let $\sigma^2 = \|\sum_{...
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Given a large random matrix, how to prove that every large submatrix whose range contains a large ball?
Context. Studing a problem in machine-learning, I'm led to consider the following problem in RMT...
Definition. Given positive integers $m$ and $n$ and positive real numbers $c_1$ and $c_2$, let's ...
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Law of large numbers and Central Limit Theorem for eigenvalues of perturbed matrices
I'm looking for results where perturbation by iid random entries to a matrix will result in convergence of the eigenvalues to the original eigenvalues. More precisely,
Let $ \forall n \in \mathbb{N},...
- 1,467
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Tracy Widom type results for asymptotic distribution of the $k$-th largest eigenvalue of the sample covariance when $n, p \to \infty$?
Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here ...
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Spectrum of large random asymmetric matrices with correlation
Background:
In their paper, Sommers Crisanti Sompolinsky and Stein derive the spectral distribution of large random matrices $\mathbf{J}$ by studying the following integral:
\begin{equation}
I=\left[\...
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Eigenvalue distribution of a band matrix
Let $\mathbf M_i$ be rectangular matrices of dimensions $N_{i-1}\times N_i$. We assume that their entries are random, with zero mean and variance $\sigma_i^2$.
For some positive integer $k$, I define ...
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101
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Controlling the rank of a Matrix product
Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form
$$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\...
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Hutchinson-type algorithm for efficient computation of trace of inverse of non symmetric matrix
Let $A$ be an invertible $N$-by-$N$ matrix, for some large $N$ (say $N = 10^6$). Suppose the only thing we know how to do is apply $A$ to a vector, i.e compute matrix-vector products $Az$.
Question....
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What is the expected inverse of 1 plus a Wishart distribution?
Let $$ X \sim \mathcal{W}_{n} (n, \Sigma) \; \;$$
Where $\mathcal{W}$ denotes the Wishart distribution and $\Sigma \in \mathbb{R}^{q \times q}$ is the corresponding scale matrix (symmetric, positive ...
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Concentration or distribution of the scaled $l_p$ norm of a correlation matrix
Background:
Among Hermitan random matrices, correlation matrix has a lot of applications in statistics. People have studied the "empirical spectral distribution (ESD)" of a correlation matrix, the ...
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Linear Independence of random binary vectors
Suppose we have $Y_1, \ldots, Y_n \in \mathbb{R}^m$, $n$ independent random vectors ($m \geq n$), where the entries of each $Y_i$ are i.i.d. Bernoulli random variables taking the values $\{0, 1\}$ ...
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Row Space of Random Matrix
Say we have a Gaussian distribution in $\mathbb{R}^p$, $x \sim N(0, \Sigma)$, and a set of $n < p$ samples from this distribution, $X \in \mathbb{R}^{n \times p}$. The projector onto the row-space ...
- 509
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341
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Riemann Explicit Formula
I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula:
$$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)...
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The condition for random positive matrice integration
For a $k \times k$ positive matrix $V=(v_{ij}), $ write $V=\Gamma'D\Gamma,$ where $D=diag(d_1,d_2,\ldots,d_k)$ with $d_1>d_2>\cdots>d_k,$ and $\Gamma$ is orthogonal matrix. From the result of ...
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128
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expected value of powers of a gaussian matrix
Let $Z$ be a fixed $d \times d$ matrix and let $G$ be a random $d \times d$ matrix with each entry i.i.d. $N(0, 1)$.
Is it true that:
$$\mathrm{Tr}(\mathbb{E}_G[ (Z^T + G^T)^\ell (Z + G)^{\ell-k-1}...
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135
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Error correcting codes via random matrices: How close to the Shannon bound?
I have a vague and probably rather naive question on error correcting codes.
Suppose we want to encode binary vectors of length $k$ as binary vectors of length $n$ in such a way that differences of ...
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270
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Expected value of inverse of complex non-central Wishart matrix
I have a matrix $W$ that abides a complex non-central Wishart distribution.
My question is what the expectation of the inverse is, i.e., how to compute
$$\mathbb{E}(W^{-1}).$$
I have tried to read up ...
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175
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Matrix Bernstein for spherical random variables
Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
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What is the distribution of engenvalues of covariance matrix when the covariance has some block diagonal structure
Let's say we have a matrix $X \in \mathbb R^{n\times p}$, where $X_{i,j}$ sampled from a Gaussian $N(\mu, \sigma^2)$, we use $\Phi$ to denote $\{\mu,\sigma\}$ for simplicity.
Now, we sample $m$ ...
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Conditonal convergence implies convergence?
Note : All measures below are probability measures.
Let $\mu_n(X,Y)$ be a random probability measure on $\mathbb C$ depending on two random variables X and Y with values in $\mathbb{R}^N$.
Actually,...
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A "conjectured" concentration inequality for operators, probably related with random matrix theory
I am working on some open problem. And I have reduced the original problem to the "conjecture" (actually I am not familiar with random matrix theory or other fields that may have such a result) as ...
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126
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Dixon-Anderson-Selberg integral variant
I am trying to evaluate or at least obtains bounds for the following integral for $0<\gamma^{2}<2$
$$ \int_{[0,1]^{2n}}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2-\...
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Empirical approaches to validate observational bounds on minimum gap between least eigenvalues of $n \times n$ correlation matrix and its submatrices
Let
$\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$.
$\Sigma_i'$ be an $(n-1) \times (n-1)$ submatrix of $\Sigma$ obtained by eliminating the $i$-th row ...
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Trivial zeros of $\zeta$ from limit characteristic functions of random matrices
Reviewing some of the literature on random matrices I have seen several studies and results on characteristic polynomials of random matrices, usually of fixed size/degree $N$. Zeros then are either on ...
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Distribution of top singular vector of Bernoulli ensemble
Consider the distribution given by taking the top singular vector of a matrix whose entries are equally likely to be +1 or -1.
This is not well-defined for matrices where the subspace achieving the ...
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Distribution of maximum minor of a random matrix with one special column
Given $m,n,\ell\in\Bbb N$ and $\beta\in(0,1)$ consider the uniformly picked random matrix $A\in\Bbb Z^{n\times (n+1)}$ with $0\neq|\mathsf{det}(A^\circ)|\leq m^{\frac 1\ell}$ where $A^\circ$ is the ...
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minimum eigenvalue of Katri-Rao product of two Gaussian matrices
Let $\mathbf{A}\in\mathbb{R}^{k\times n}$ and $\mathbf{B}\in\mathbb{R}^{d\times n}$ be independent matrices with i.i.d. $\mathcal{N}(0,1)$ entries. I'm interested in lower bounding the minimum ...
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psd condition for matrix completion
The nuclear norm minimization for the matrix completion problem is given by
\begin{align}
\textrm{minimize } \quad &\|X\|_{*}\\
\textrm{subject to } \quad & X_{ij}=M_{ij} \quad \forall (i,j)...
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One-sided Talagrand concentration inequality for empirical processes
Let $\mathcal{F}$ denote a function class. A classic result by Talagrand states that
\begin{align*}
\mathbb{P}\bigg\{\sup_{f\in\mathcal{F}}\big|\sum_{i=1}^nf(X_i)-\mathbb{E}\big[\sum_{i=1}^nf(X_i)\...
- 859
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On the numerical range of non-self adjoint Gaussian matrix
For a complex $n \times n$ matrix $A$, its numerical range is the set
$$W(A) = \left\{\mathbf{x}^*A\mathbf{x} \mid \mathbf{x}\in\mathbb{C}^n,\ \|x\|_2=1\right\} .$$
We can further define the ...
- 1,503
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Majorizing inequality on spectral norm of product of a random and a deterministic low-rank projection
Let $P$ be a rank $k$ uniformly randomly oriented projection matrix in ${\mathbb R}^d$ -- this is constructed as $R^T(RR^T)^{-1}R$ where $R$ is a $k\times d, k<d$ random matrix with i.i.d. 0-mean ...
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Products of random permutations with fixed matrix
This question originates from an engineering problem, which I am solving. Any related references are highly appreciated.
Let $M_k(T)=\prod_{t=1}^T P_t S_k$ over some field (finite or reals), where $...
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Stochastic independence of columns of projection matrix to the rest of the columns of a random matrix
First let me describe the setting of the problem.
I have a random matrix $A\in \mathbb{R}^{m\times n},\ (m<n)$ with $a_{ij}\sim \mathcal{N}(0,I)$ i.i.d. Let there be a given set of $K (K<m)$ ...
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Projection of a ray onto a random polytope
Suppose $P$ is a polytope formed by $p$ (general) random planes in $\mathbb{R}^n$. We assume $p \asymp n$ and $P$ has a diameter $O(\sqrt{n})$. For any $x \in \mathbb{R}^n$, denote by $\operatorname{...
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Bound the expectation of trace norm of random Hermitian matrix
Suppose $H_i$ are traceless $d\times d$ Hermitians, $X_i$ are Standard normal distribution for $1\leq i\leq d^2$.
We would like to bound the following expectation on the trace norm
$\mathbb{E}|\sum_{...
- 1,503
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112
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Tools to bound the singular values of a finite sum of random matrices from below?
Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
- 111
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57
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Inverse of the covariance of the estimate of a covariance
I have a covariance matrix, $V_{ij}$, which (for reasons that aren't important) I'm going to call the visibilities. I have an estimator for the visibilities $\hat V_{ij}$, and I've derived that the ...
- 211
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225
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Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?
By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the $n!...
- 1,893
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46
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the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?
Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance matrix,...
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260
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Distribution of the Gram Matrices
Let $\mathbf{X}$ be an $m\times m$ random matrix full rank matrix, having the density function $f_{\mathbf{X}}(X)$. Also, let $\mathbf{W}$ be a deterministic $k\times m$ matrix of rank $k$ and $k<m$...
- 141
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132
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Eigen value distribution of autocorrelated Wishart matrix
Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
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93
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Random Schrödinger operators with asymmetric Lifshitz tails?
For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ ...
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