Questions tagged [random-matrix]

A random matrix is a matrix whose entries consist of random variables from some specified distribution. Random matrices have many modern applications in physics, finance, statistics and numerical analysis.

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significance of PCA with EWMA

I am evaluating a PCA on a set of 45 financial time series where EWMA (exponentially weighted moving averages) has been applied. A small intro. An EWMA statistic has a form $$ \hat{y}_t = \sum_s \...
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On distributions over orthonormal sets: existing families, construction, and simulation

Have families of distributions over orthonormal sets been defined and studied in the literature? What are a couple examples and/or references? Are there known methods for constructing distributions ...
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Distribution of a product of a matrix with a random matrix

Suppose we have the matrices $Z\in \mathbb{R}^{n\times n}$ and $X\in \mathbb{R}^{n\times d}$, such that each row $x_i\in\mathbb{R}^d$ is drawn i.i.d from a $N(0,\Sigma_{d\times d})$ distribution. ...
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Probability of norm of dependent vector elements

Suppose $A^T x=0$ holds for matrix $A \in R^{n \times m}$ and vector $x \in R^n$. Define vector y as: $y = (S A)^T Sx = \left[ \begin{array}{c} (A_{*1})^T S^T S x \\ \vdots \\ (A_{*m})^T S^T S x \end{...
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Effect of random matrices on the product of two perpendicular matrices

For two non-square perpendicular matrices $X$ and $Y$ (i.e., $X^T Y = 0$), what happens when we bring a random matrix $A \sim \mathcal{N}(0,\sigma^2)$ (with i.i.d. elements) to the inner product, as ...
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Concentration of top eigenvectors

Let $M$ be a $d \times d$ symmetric matrix with rank $k < d$, write $M = U \Lambda U^T$. Define $\hat{M} = M + Z$, where $z_{ij} \sim N(0, 1 /n )$. Suppose we try to estimate $U$ by taking the top $...
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How to check if my matrices are 'random' using Marchenko-Pastur?

For each $n=1,2,\dots,100$, I have an $r \times s$ integer-valued matrix $X(n)$ of increasing dimension as $n$ increases. That is, the number of rows $r$ and columns $s$ increases as $n$ increases. ...
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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 $\...
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Random matrix theory impact on covariance matrix analysis

Framework: From RMT, eigenvalues have a semicircle distribution for symmetric matrices each with i.i.d normally distributed entries as the size of the matrix grows. The restrictions on i.i.d have ...
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Reference request: (spectral) convergence rate of sample covariance matrix with fixed dimension $p$

I am looking for a reference on convergence of sample covariance matrix (in some reasonable sense) when the dimension $p$ is fixed, but the number of samples $n$ goes to infinity. The ideal result I ...
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PCA: inference on the proportion of explained variance, in a large p setting

I am interested in doing inference on the proportion of total variance explained by the first principal component, for a PCA based on the correlation matrix R. I want to know the (asymptotic) ...
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Statistical distance between two matrices

The statistical distance between two probability distributions can be measured with $f$-divergences such as the KL-divergence. The statistical distance between two clusters can be measured with ...
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generation of random covariane matrix

in the latest book by Marcos Lopez de Prado, he provides sample code for generating a random variance-covariance matrix. He starts by generating a rectangular dataset with fewer observations than ...
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Does $E(XX^{\top})$ being full rank imply $E(XX^{\top}\mathbf{1}(Y\in A))$ being full rank?

suppose $X=\begin{bmatrix}X_{1}\\X_{2}\end{bmatrix}$ is a discrete random vector with finite support, and $Y$ is a continuous random variable with finite support $[a,b]$, and $A$ is a subset of $[a,b]$...
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(Co)Variance of a random matrix

The expected value $\mathbb{E}[\mathbf{x}]$ of a random vector $\mathbf{x} \in \mathbb{R}^{n \times 1}$ is the vector of the expected values of each individual random variable $\mathbf{x}$ contains. ...
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Variance of Random Vector in the Circular Orthogonal Ensemble

Let $x$ be a (uniformly) randomly chosen column of a random orthogonal matrix (of size $K$ x $K$) distributed according to Haar measure. What is $\mathbb{E}[x]$, $\mathbb{E}[x x^T]$, $Cov(x, x)$, and $...
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Showing a useful result for Wisharts and Multivariate Beta random matrices

Let $\mathbf{A} \sim \text{Wishart}_m\left(k_a,\mathbf{V} \right)$ and $\mathbf{B} \sim \text{Wishart}_m\left(k_b,\mathbf{V} \right)$ be two full rank Wishart random matrices. Define $$ \mathbf{S} = \...
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Random matrix theory and research - a lot like doing linear algebra?

I've been searching for a subfield of research to get into and wonder whether random matrix theory could suit me well; it seems like it does, because the stuff I read, and the seminars that I watch ...
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Minimax equivalence of Matrix Norm

In a proof of random matrix theory, the author makes use of the following equivalence: \begin{equation} \inf_{v\in V(r)}\lVert Xv \rVert_2 = \inf_{v\in V(r)} \sup_{u \in S^{n-1}}u^TXv \end{equation} ...
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Factorizing a matrix of distributions [closed]

Let's say we have a matrix $X \in \mathbb{R}^{m \times n}$, then the (R-truncated) SVD allows to approximate: $X_{i,j} \approx \sum\limits_{r=1}^{R} \sigma_r \times U_{r,i} \times V_{r, j}$ Now I ...
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marchenko pastur for Correlation

It has been suggested to me that if I construct a covariance or correlation matrix using factor model then I can use the Marchenko-Pastur distribution to highlight significant correlations (or ...
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How to estimate the largest eigenvalue of a correlation matrix from one observation of underlying data matrix?

Suppose that I have $N$ time series $x_{1t},x_{2t},\dots,x_{Nt},$, that are correlated with each other. A $N\times N$ correlation matrix is $R=\rho_{ij}$. It can be represented with eigen value ...
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Marginal Distribution of Matrix Normal with Two Inverse Wisharts

Say I have a Matrix-Normal distribution and two Inverse Wishart Distributions $$X \sim MN_{p\times n}(0, \Sigma, \Omega)$$ $$ \Sigma \sim IW(a, A) $$ $$ \Omega \sim IW(b, B)$$ where $a$ and $b$ are ...
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Are there random matrix results like Marcenko-Pastur, but for CCA?

The Marcenko-Pastur law is about asymptotic distributions of eigenvalues. It starts from a simple null model (iid zero-mean Gaussian entries) and derives a distribution for the spectrum. In PCA, this ...
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How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution?

Problem Given $X \in \mathbb{R}^{n \times n}$ where $X_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma_{ij}^2 I)$ Find the eigenvalue distribution using whatever you can. Background In my field, I have a ...
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5 votes
1 answer
262 views

Transformation of Inverse Wishart

Let $\Sigma$ be an $p\times p$ dimensional covariance matrix that is distributed Inverse Wishart with degrees of freedom $\nu$ and Prior scale matrix $\Psi$ such that we write $\Sigma \sim W^{-1}(\nu, ...
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Random rotation of a set of distinct points in $R^n$

Consider a set $\{\mathbf{X}_1,\cdots , \mathbf{X}_M\}$ of distinct points in $\mathbb{R}^n$ with $M$ finite. The $M$ values of the $i$-th coordinate do not all have to be dinstinct. For example, in $\...
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Expectation of a Matrix Raised to a Power

I have been able to find little literature related to the topic online (e.g., A note on the Expected Value of an Inverse Matrix such that E($\frac{1}{X})$ $\geq$ ($\frac{1}{E(X)}$), though, vaguely, I ...
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1 answer
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Rank of N x D vs D x N matrices

If $X$ is a random $N \times D$ matrix where $N > D$, then why is the rank of X - mean(X, 1) $D$ while the rank of ...
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4 votes
0 answers
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Distribution of $\mathbf{A}\mathbf{X}$?

Let $\mathbf{A}$ be an $m\times n$ random matrix with entries $A_{ij}$ being jointly Gaussian. Suppose all of these variables are independent of the random vector $\mathbf{X} = (X_1,\ldots,X_n)^\top$ ...
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conjugate prior (and posterior) for Matrix Variate Distributions

I was wondering what is the conjugate prior for Matrix Variate Distributions (e.g., unknown mean, known variance matrices), and what's the corresponding posterior? Is there analytical solution?
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Distribution of eigenvalues of a random matrix

Given a random symmetric matrix A whose entries are Poisson distributed, can anything be said about the distribution of A's eigenvalues? Would be great if someone could link a paper citing such a ...
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33 votes
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If I generate a random symmetric matrix, what's the chance it is positive definite?

I got a strange question when I was experimenting some convex optimizations. The question is: Suppose I randomly (say standard normal distribution) generate a $N \times N$ symmetric matrix, (for ...
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Variance of Random Matrix

Let's consider independent random vectors $\hat{\boldsymbol\theta}_i$, $i = 1, \dots, m$, which are all unbiased for $\boldsymbol\theta$ and that $$\mathbb{E}\left[\left(\hat{\boldsymbol\theta}_i - ...
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3 votes
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Law of Total Expectation and Law of Total Variance for Matrices

Does anyone have a reference or proof for the LTE and LTV for matrices? I am defining the unconditional variance for matrices in the usual way: $$ \operatorname{Var}_{m}(M) \overset{\text{def}}{=} \...
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3 votes
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What is popular choice for modelling distribution of orthogonal matrices

In Bayesian statistics, instead of considering a variable to be fixed and use MLE to infer that value, we put a prior distribution over that variable. Now consider an orthogonal matrix $W \in R^{d \...
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Eigenvalue order of magnitude for Wishart random matrix

If we have a $P\times N$ matrix $\mathbf{A}$ whose elements $A_i$ are samples (in this case, P samples) from a multivariate gaussian distribution in an $N$ dimensional space, we can define the Wishart ...
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Why did statisticians define random matrices?

I studied mathematics a decade ago, so I have a math and stats background, but this question is killing me. This question is still a bit philosophical to me. Why did statisticians develop all sort of ...
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Product of two random Gaussian matrices - orthant probability

In my studies of random matrices, I recently came across this challenge: Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., ...
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Marginal distribution of distribution on the Unitary group [closed]

I am trying to understand the behavior of distributions over the Unitary group (i.e. the set of square matrices $P$ such that $P^tP = I_d$), or in general distribution over the Stiefel manifold (set ...
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4 votes
1 answer
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Does this method uniformly sample 3x3 rotation matrices?

Let $\textbf{X}=(X_1,X_2,X_3)$ be a random unit vector in $\mathbb{R}^3$ such that $\textbf{X}$ is uniformly distributed on the unit sphere $S^2$. Next, let $\textbf{Y}=(Y_1,Y_2,Y_3)$ be a random unit ...
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8 votes
0 answers
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Intuitive explanation for Marchenko-Pastur law

I am looking for an intuitive reasoning behind the Marchenko Pastur law, which is described as a law of large numbers analog for random matrices. I know the law gives the probability density function ...
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Hypothesis testing for matrices of RVs

I looking for the correct statistical test for comparing matrices of random variables - my IV being a categorical variable and my DVs being n by n matrices of random variables (which are themselves ...
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2 votes
2 answers
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Write $\mathop {\mathbb E}[(X AX^h)]$ in function of $\mathop {\mathbb E}[(X X^h)]$?

Suppose $X$ is an $i \times j$ random matrix. In addition, $X$ has complex i.i.d. normal entries with $0$ as mean. We define $A$ (of dimension $j \times j$) as a deterministic matrix. Is it ...
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5 votes
1 answer
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How to check if a distribution has undefined variance?

How can I determine if experimental data comes from a distribution where the variance is undefined (e.g. the Cauchy distribution)? I honestly have no idea how to attack this problem in a sensible way,...
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4 votes
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Convergence of eigenvectors and eigenvalues of matrix that converges

For each random variable $X=x$, there is a symmetric positive definite matrix $M(x)$. Suppose there is a set of samples of random matrix $M_1,M_2,...,M_n$, where each $M_i$ is calculated based on the ...
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4 votes
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Parametric distributions over orthogonal projection matrices?

Consider the set of rank $m$ orthogonal projection matrices in $\mathbb{R}^{d\times d}$, for some $m<d$. With appropriate measure-theoretic considerations, one can define multiple distributions on ...
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1 answer
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Smoothing Kernel Preventing Simulation of Semi-Circle Law of Random Matrices

In trying to understand the properties of random matrices in the book "Plane Answers to Complex Questions" by R.Christensen, I came across the Semi-Circle Law, and tried to reproduce it with ...
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4 votes
1 answer
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Eigenvectors of a Wishart matrix

I have been trying to find a good source (or clarifications) to help me understand this point. I am very new to random matrix theory so any pointers will be appreciated. Here is what I think I have ...
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2 votes
2 answers
278 views

Spectral norm of a sparse Gaussian matrix

Suppose $G$ is an $m \times n$ matrix such that each entry of $G$ is a standard normal variable. We know that the spectral norm of $G$ scales as $\sqrt m + \sqrt n$. Now, given a set of indices $S$ ...
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