This is regarding the compressed sensing problem that I recently learned about. In the problem $y=X\beta$ where $y\in\mathbb{R}^{n}$, $X\in\mathbb{R}^{n\times p}$, and $\beta\in\mathbb{R}^{p}$, it is known that a random matrix $X$ where $X\sim_{\text{iid}}N(0,1/n)$ satisfy the Restricted Isoperimetric Property (RIP) condition with high probability.
My questions:
- Is it true that if $X_{ij}\sim_{\text{iid}}\text{Bernoulli}(\alpha)$ where $\alpha$ is a fixed constant in $(0,1)$, then $X$ still satisfies RIP with high probability? This paper looked at symmetric Bernoulli matrices where entries are in $\{-1,+1\}$ instead of $\{0,1\}$ which is what I am looking at.
- We have a matrix constructed from a diagonal of matrices: $$ X=\text{diag}(A,\dots,A) = \begin{bmatrix} A & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & A \end{bmatrix}, $$ where $A\in\{0,1\}^{L\times L}$ and entries of $A$ are from $\text{Bernoulli}(\alpha)$ with fixed constant $\alpha$ in $(0,1)$. The scaling of $L$ is $L\in\Theta(p)$, $n/p=\Theta(1)$, $L<n,p$. Does $X$ still satisfy the RIP condition with high probability?
Thanks.