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


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

  • $\begingroup$ Do you know what $\|\mathbf{x}\|_1$ means? $\endgroup$ – BlackMath Apr 1 at 14:46
  • 2
    $\begingroup$ Your $ \hat{x} $ isn't well defined. Are we looking for argument of which function? $\endgroup$ – Royi Apr 1 at 16:33
  • $\begingroup$ $\|x\|_1$ is the $\ell_1$ norm. $\endgroup$ – Shashank Ranjan Apr 1 at 17:30
  • $\begingroup$ What is $l_1$ norm? $\endgroup$ – BlackMath Apr 1 at 17:32
  • $\begingroup$ argument of the function $\hat{x}$, which is called the basis pursuit. $\endgroup$ – Shashank Ranjan Apr 1 at 17:33

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