Restricted isometry property

In compressed sensing, Restricted isometry property (RIP) is a widely used condition on the linear measurement operator which can guarantee robust recovery of the sparse signals via $\ell_1$ minimization. Basically, an operator satisfying the restricted isometry property will approximately preserve the length of all signals up to a certain sparsity. Many recovery results on compressed sensing are based on RIP and in particular matrices with independent and identically distributed subgaussian entries can be guaranteed to have good RIP constants with an optimal number of measurements $O(klog(n/k))$ where $k$ is sparsity and $n$ is the dimension of the sparse vector.

Definition (Restricted Isometry Constant)
Restricted isometry constant $\delta_k$ of a matrix $A$ is the smallest number so that for all $x\in\mathbb{R}^n$ with sparsity at most $k$ we have \begin{equation} (1-\delta_k)\|x\|_{\ell_2}^2\leq \|Ax\|_{\ell_2}^2\leq (1+\delta_k)\|x\|_{\ell_2}^2 \end{equation}

Theorem
\begin{equation} \label{ell1min} \min \|x\|_{\ell_1}\text{  subject to   }y_0=Ax \end{equation} Assume $y_0=Ax_0$, $x_0$ is at most $k$ sparse and $\delta_{2k}<\sqrt{2}-1$ for measurement $A$. Then $x_0$ is the unique minimizer of the basis pursuit given in \ref{ell1min}.