Editing Open Problems:24
Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.
The edit can be undone.
Please check the comparison below to verify that this is what you want to do, and then save the changes below to finish undoing the edit.
| Latest revision | Your text | ||
| Line 5: | Line 5: | ||
We say that a vector $v \in \mathbb R^n$ is ''$k$-sparse'' for some $k \in \{0,\ldots,n\}$ if there are no more than $k$ non-zero coordinates in $v$. The goal in the problem being considered is to | We say that a vector $v \in \mathbb R^n$ is ''$k$-sparse'' for some $k \in \{0,\ldots,n\}$ if there are no more than $k$ non-zero coordinates in $v$. The goal in the problem being considered is to | ||
design an $m \times n$ matrix $A$ such that for any $x \in R^n$, one can recover from | design an $m \times n$ matrix $A$ such that for any $x \in R^n$, one can recover from | ||
| β | $Ax$ a vector $x^* \in \mathbb R^n$ that satisfies the following “$L_2/L_1$” | + | $Ax$ a vector $x^* \in \mathbb R^n$ that satisfies the following “$L_2$/$L_1$” |
approximation guarantee:\[ \left\|x^*-x\right\|_2 \leq \min_{k\text{-sparse}\,x'\in\mathbb R^n} \frac{C}{\sqrt{k}} \left\|x'-x\right\|_1,\]where $C>0$ is a constant. | approximation guarantee:\[ \left\|x^*-x\right\|_2 \leq \min_{k\text{-sparse}\,x'\in\mathbb R^n} \frac{C}{\sqrt{k}} \left\|x'-x\right\|_1,\]where $C>0$ is a constant. | ||
| Line 14: | Line 14: | ||
It is known that one can achieve ''either'' (a) ''or'' (b) (see, | It is known that one can achieve ''either'' (a) ''or'' (b) (see, | ||
e.g., {{cite|NeedellT-10}}). It is also possible to achieve both (a) and (b), but with | e.g., {{cite|NeedellT-10}}). It is also possible to achieve both (a) and (b), but with | ||
| β | a different “$L_1/L_1$” approximation guarantee, where $\|x^*-x\|_1 \leq \min_{k\text{-sparse}\,x'} C \|x'-x\|_1$ {{cite|IndykR-08|BerindeIR-08}}. | + | a different “$L_1$/$L_1$” approximation guarantee, where $\|x^*-x\|_1 \leq \min_{k\text{-sparse}\,x'} C \|x'-x\|_1$ {{cite|IndykR-08|BerindeIR-08}}. |
