Skip to main content

All Questions

Filter by
Sorted by
Tagged with
1 vote
0 answers
80 views

Inequality involving random vectors and absolute values

Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
0 votes
1 answer
158 views

Techniques for bounding the operator norm of the expectation of random matrix?

Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix $$ A = I_n - uu^T. $$ Question: What techniques are available to provide (reasonably ...
0 votes
1 answer
115 views

Approximation for an expectation expression

Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
5 votes
1 answer
274 views

Precise probability that $m$ random vectors in $n$ dimensional space are nearly-orthogonal

Consider $m$ vectors $v_1,\dots,v_m$ in $\mathbb R^n$, drawn uniformly and independetly from unit sphere. It is pretty straightforward from Chebyshev inequality that $$ \mathrm P (\forall i\ne j \ |...
1 vote
1 answer
607 views

Hanson-Wright inequality with random matrix

I'm interested in bounding the tail probabilities of a quadratic form $x^t A x$ where $x\in \mathbb{R}^n$ is a sub-Gaussian vector with independent entries. $A\in \mathbb{R}^{n\times n}$ is a matrix. ...
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 ...
1 vote
0 answers
110 views

Tail bound without independence

Suppose $X_i , X_j\in \mathbb{R}^d$ are gaussian vectors and $A$ is an $n\times n$ symmetric PSD matrix where $A_{ij} = f(\|X_i-X_j\|_2), \quad i,j\in 1,\ldots,n\;$ for some non-negative Lipschitz ...
1 vote
0 answers
282 views

Strict monotonicity of conditional variances

Let $K \geq 2$ be a positive integer and $C$ be any $K \times K$ non-singular matrix (if necessary, can assume that all $K$ rows of $C$ are needed to span the coordinate row vector $e_1'$). For ...
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 ...
6 votes
1 answer
239 views

Positive semidefinite ordering for covariance matrices

Suppose that X and Z are matrices with the same number of rows. Let $$ D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & ...
3 votes
0 answers
158 views

Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added. This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...