All Questions
10 questions
14
votes
1
answer
449
views
References for reasoning about the spectrum of a convex body?
By "spectrum of a convex body", I mean: start with a convex body $B$ in $\mathbb{R}^d$, then consider the corresponding $d \times d$ covariance matrix resulting from a uniform distribution over $B$ -- ...
- 143
4
votes
1
answer
1k
views
Expected value of the spectral norm of a Wishart matrix?
Let $x_1,\dots,x_n$ be i.i.d. drawn from $N(0,I_{p\times p})$. Consider the sample covariance matrix $W(n,p)=\frac 1n \sum_{i=1}^n x_ix_i^T$, a Wishart matrix.
For fixed $n,p$, what is the expected ...
- 947
2
votes
1
answer
2k
views
Bounds on the eigenvalues of the covariance matrix of a sub-Gaussian vector
Suppose that $\boldsymbol{x}\in\mathbb{R}^n$ is subgaussian random vector of variance proxy $\sigma^2$, i.e.,
$$\forall \boldsymbol{\alpha}\in\mathbb{R}^n: \quad \quad \mathbb{E}\left[ \exp\right(\...
- 127
2
votes
3
answers
999
views
Sum of Square of the Eigenvalues of Wishart Matrix
Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
- 1,096
2
votes
1
answer
280
views
The effect of random projections on matrices
Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$.
Suppose $\frac{A+...
- 1,495
2
votes
2
answers
144
views
Spectrum of finite-band random matrices?
Let
$X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that :
$$ \begin{cases}
&X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\
& X_{ij} \sim P_X \quad \text{otherwise}
\end{cases}$$
And ...
- 245
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
1
vote
2
answers
892
views
Concentration of matrix norms under random projection.
Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables.
Are there cases where one can been able to quantify $P_G ...
- 617
1
vote
1
answer
206
views
Moment generating function of spectral norm of iid N(0,1) data matrix
Let $W^{p\times p}$ be a normal data matrix with $W_{ij}$ i.i.d. $N(0,1)$. Are there any results on the evaluation, or upper bound for the Moment Generating Function of the spectral norm of W, that is,...
- 113
0
votes
1
answer
622
views
Is there any way to compare between diagonals of a resolvent and a Cauchy transform?
Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...
- 617