All Questions
28 questions with no upvoted or accepted answers
36
votes
0
answers
2k
views
Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices
In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
- 188k
8
votes
0
answers
232
views
Decay of orthogonal contributions in a random set of vectors
Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\...
- 2,252
6
votes
0
answers
279
views
Estimating $E[\operatorname{Tr}(ABABBA..)]$ for random shuffling of $A,B$?
How can I estimate the following value where $A,B$ are $d\times d$ matrices and expectation is taken over all random permutations of the product?
$$E_\text{shuffle}[\operatorname{Tr}\underbrace{AA\...
- 2,252
5
votes
0
answers
235
views
Riemann theta function inequality for a class of large random matrices
The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
- 686
4
votes
0
answers
463
views
The distribution of the elements of an eigenvector of random matrices
Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...
- 39
3
votes
0
answers
58
views
Projection onto column space perturbed by Gaussian noise
Suppose we have a matrix $X\in\mathbb{R}^{m\times n}$ (with $n \le m$) with iid standard Gaussian entries, and suppose we have noise matrix $W\in\mathbb{R}^{m\times n}$ with iid Gaussian entries, but ...
- 141
3
votes
0
answers
78
views
Using Kac-Rice formula to count average number of sub-regions carved out by $n$ random hyper-planes
Context. This is the first in a set of tiny pieces of a problem I've formulated to help me measure the "complexity" of certain piecewise linear functions. Thanks in advance for your help and patience.
...
- 6,853
3
votes
0
answers
77
views
A concentration problem of product of matrices
Let $A$ be an $n \times m$ matrix with non-negative entries and $B \in \mathbb{R^{n\times n}_{\geq 0}}, C \in \mathbb{R^{m\times m}_{\geq 0}}$ be random matrices where B and C are both symmetric and ...
- 195
3
votes
0
answers
202
views
Difficult Gaussian-sum inequality for large random Bernoulli-Toeplitz matrices
I have come across the following problem in an attempt to prove an entropy bound for large random Bernoulli-Toeplitz matrices (Conjecture 1 on p. 16 of this preprint by Clifford et al. 2015), which is ...
- 686
2
votes
0
answers
269
views
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 \...
- 121
2
votes
0
answers
132
views
Limiting PDF of the eigenvalue of random Gaussian matrix
It has been proven that the CDF of the eigenvalue distribution of random Gaussian matrix converges to a uniform disk circular law. Is it true for the PDF of the limiting eigenvalue distribution? In ...
- 133
2
votes
0
answers
116
views
Smallest singular value distribution
Let $G_\mathbb{R}\in\mathbb{R}^{n\times n}$ and $G_\mathbb{C}\in\mathbb{C}^{n\times n}$ denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian ...
- 83
2
votes
0
answers
102
views
Eigenvalue distribution for a real-valued random matrix with correlated Gaussian entries
I'm working on an application where I would greatly benefit from knowing the distributions of the eigenvalues of a real-valued random matrix whose elements can be assumed to be Gaussian, but where I ...
- 121
2
votes
0
answers
367
views
Eigenvalue perturbation of a symmetric matrix by a random orthogonal projection
Given fixed real symmetric $D\in\mathbb{R}^{n\times n}$ with $n$ distinct eigenvalues, let $U$ be a random orthogonal matrix selected uniformly from the space of $n\times n$ orthogonal matrices, and ...
- 570
2
votes
0
answers
266
views
Smallest Singular Value of a Random Matrix with Dependent Entries
Overview
I am trying to bound from below the smallest singular value $\sigma_{n}$ of a sequence of symmetric $n$ by $n$ random matrices $M_{n}$ with dependant entries. In particular, I would like to ...
- 98
2
votes
0
answers
1k
views
Random matrices whose limit gives exact Wigner surmise
Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write $\rho^W_0(s)=\...
- 4,952
2
votes
2
answers
215
views
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 ...
- 287
1
vote
0
answers
66
views
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:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\...
- 284
1
vote
0
answers
1k
views
Linear Independence of random binary vectors
Suppose we have $Y_1, \ldots, Y_n \in \mathbb{R}^m$, $n$ independent random vectors ($m \geq n$), where the entries of each $Y_i$ are i.i.d. Bernoulli random variables taking the values $\{0, 1\}$ ...
- 111
1
vote
0
answers
128
views
expected value of powers of a gaussian matrix
Let $Z$ be a fixed $d \times d$ matrix and let $G$ be a random $d \times d$ matrix with each entry i.i.d. $N(0, 1)$.
Is it true that:
$$\mathrm{Tr}(\mathbb{E}_G[ (Z^T + G^T)^\ell (Z + G)^{\ell-k-1}...
- 199
1
vote
0
answers
60
views
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 $K (K<m)$ ...
0
votes
0
answers
92
views
Linear independence of Wishart matrices
Let $W\sim W_n(I,d)$ be a real Wishart matrix of an identity covariance matrix and $d$ degrees of freedom, i.e., $W=XX^T$ for $X$ being an $n\times d$ matrix whose entries are i.i.d sampled from a ...
- 123
0
votes
0
answers
45
views
On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,826
0
votes
0
answers
47
views
"Probability" for a partitioned matrix to be singular
Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix
$$
M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
- 2,712
0
votes
0
answers
132
views
Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix
Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
- 109
0
votes
0
answers
64
views
Probability of collision of sums of vectors
Let $S_1$ and $S_2$ be sets of vectors from $\mathbb{R}^d$ that are distinct and let $\sigma(\cdot)$ be a non-linearity, e.g., a componentwise sigmoid function.
Does there exist a random matrix $R \...
- 321
0
votes
0
answers
166
views
Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?
Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...
- 87
-1
votes
1
answer
68
views
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 ...
- 25