Suppose we want to recover an input vector $\displaystyle f \in \mathbb{R}^n $ from corrupted measurements $\displaystyle y = Af + \epsilon $. Note that $\displaystyle A $ is an $\displaystyle m \times n $ matrix and $\displaystyle e $ is a matrix of unknown errors. So then we can recover $\displaystyle f $ exactly from the data $\displaystyle y $ by first identifying the error and subtracting it off. Then we need something that annihilates $\displaystyle A $ so that we can get $\displaystyle f $. In other words, we want a function such that:

$\displaystyle f(Af+ \epsilon) = f(Af)+f(\epsilon) = f(\epsilon) $.

In other words, we are looking at the kernel of $\displaystyle Af $. So once we identity the error term...can we recover $\displaystyle f $ exactly? Because given $\displaystyle Af+ \epsilon $ and $\displaystyle \epsilon $, we want to get $\displaystyle f $. So ultimately we want to do the following:

- Identify and subtract error term: $\displaystyle (Af+ \epsilon)-\epsilon $
- Now we are left with $\displaystyle Af $. So we want something that annihilates $\displaystyle A $.

Is this possible?