# What is the error rate in compressed sensing?

Let $$x \in \mathbb{R}^n$$ be a $$k$$-sparse vector. Given $$A \in \mathbb{R}^{m \times n}$$, we have a measurement vector $$y$$ given by

$$y=Ax$$

Let $$\hat{x}$$ be defined as follows

$$\hat{x}:=\arg\min_{z\in\mathbb{R}^n}\|z\|_1,\ \mbox{s.t}\ Az=y$$

Then, which of these two would be called error rate?

1. $$\|\hat{x}-x\|_1$$, or,

2. $$\dfrac{\|\hat{x}-x\|_1}{\|x\|_1}$$.

• Do you know what $\|\mathbf{x}\|_1$ means? – BlackMath Apr 1 at 14:46
• Your $\hat{x}$ isn't well defined. Are we looking for argument of which function? – Royi Apr 1 at 16:33
• $\|x\|_1$ is the $\ell_1$ norm. – Shashank Ranjan Apr 1 at 17:30
• What is $l_1$ norm? – BlackMath Apr 1 at 17:32
• argument of the function $\hat{x}$, which is called the basis pursuit. – Shashank Ranjan Apr 1 at 17:33