All Questions
Tagged with random-matrices inequalities
17 questions
3
votes
0
answers
57
views
Maximizing a Gaussian quadratic form
Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$).
Consider the map
$$
f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle,
$$
...
- 410
1
vote
0
answers
43
views
Moments on the Stiefel manifold
Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$.
Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
- 410
1
vote
0
answers
92
views
Multilinear non-commutative Khintchine inequality
Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...
- 156
0
votes
0
answers
77
views
Maximizing the trace of the resolvent of a Wishart matrix over positive unit trace matrices?
Let $G$ be a standard $d \times d$ Wishart random matrix and consider the problem of maximizing the function
$$
f(M) = \mathbb{E}\Big[\mathrm{tr}((G + M^{-1})^{-1})\Big],
$$
over the class of real ...
- 410
0
votes
1
answer
101
views
Can we show that for every $\delta>0$, there exist constants $\alpha>0, \beta>0$ so that the following inequality holds with high probability?
Consider two $n-$dimensional random vectors $u$ and $v$ uniformly distributed on the sphere. Define $X_n :=u\cdot v$. Note that as $n\to \infty$, $\sqrt{n}X_n \to N(0,1)$ as $n\to \infty$. Fix $\...
- 288
2
votes
1
answer
796
views
Can we get that $ P(N^{2/3}(\lambda_N-\lambda_{N-1})\le c)\ge 1-\epsilon$?
Following this question: Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy-Widom law?.
We know that
$$
\lim_{N\to\infty}P(N^{2/3}(\lambda_N-2)\le s_1,\dotsc,...
- 288
8
votes
1
answer
717
views
Relative Entropy and p-norm
I asked this question on StackExchange but could not get any answer, therefore, I am posting it here.
I am currently reading the book "A Dynamical Approach to Random Matrix Theory". The ...
- 371
1
vote
1
answer
210
views
Tighter upper bound for $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T W \sigma]$
Following this question I was thinking about ways to improve the upper bound and came up with the following argument. We want to find an upper bound for
\begin{equation}
\mathbb{E} [\max_{\sigma \in \{...
- 237
13
votes
2
answers
656
views
Random matrix with given singular values
Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let
$$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\...
- 1,495
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
125
views
Concentration of sums of random matrices around the mean, in the Loewner order
Recently, I have found myself interested in concentration properties of random matrices.
Specifically I would like to answer questions of the following sort
Let $\{X_i\}_{i=1}^n$ be i.i.d. copies ...
- 393
1
vote
1
answer
121
views
Probability for high mutual coherence on all subsets of a Gaussian vector set
We examine as set of independent normal vectors:
$$ \forall i \in [N]\triangleq \{1,\dots,N\}:\,\mathbf{x}_{i}\sim\mathcal{N}\left(0,\mathbf{I}_{d}\right)$$
For any $\epsilon>0$ and $K\leq N$, we ...
- 968
4
votes
1
answer
207
views
Upper bound on the number of binary matrices with small rank
I'm looking for the tightest upper bound on the number of different binary matrices $A \in {\{-1,1\}^{m \times n}}$ for which $\mathrm{rank}(A)\leq r$. I'm interested in the regime $1 \ll r \ll m \...
- 968
2
votes
2
answers
383
views
Estimating a Selberg-type integral (or a Fredholm determinant)
I am concerned with the asymptotical behavior of integrals like this for large $n$
$$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq n}(x_{j}-x_{i})^{2}\,\prod_{j=1}^{n}e^{-x_{j}^{2}}dx_{j},$$
...
- 661
6
votes
1
answer
243
views
Upper bound for a Selberg-type integral over a rectangular region
(Cross-posted from math-SE).
I am trying to estimate the values of the following integral for large $n$,
$$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq n}(x_{j}-x_{i})^{2}\,\prod_{j=1}^{n}e^{-...
- 661
1
vote
0
answers
112
views
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
1
vote
0
answers
225
views
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