| Hi-index | 754.84 |
Consider the noisy underdetermined system of linear equations: $y=Ax_0 + z$, with $A$ an $n \times N$ measurement matrix, $n , and $z \sim {\ssr N}(0,\sigma^2 {\rm I})$ a Gaussian white noise. Both $y$ and $A$ are known, both $x_0$ and $z$ are unknown, and we seek an approximation to $x_0$. When $x_0$ has few nonzeros, useful approximations are often obtained by $\ell_1$ -penalized $\ell_2$ minimization, in which the reconstruction ${\mathhat{x}^{1,\lambda}}$ solves $\min\{\Vert y - Ax\Vert_2^2/2 + \lambda \Vert x\Vert_1\}$.