In "The Restricted Isometry Property and Its Implications for Compressed Sensing" (https://statweb.stanford.edu/~candes/papers/RIP.pdf), Candes describes the Noisy Recovery problem to be as follows (problem 1): \begin{align} \text{minimize} & \hspace{1em} \| x \|_1 \\ \text{subject to} & \hspace{1em} \|y - \Phi x \|_2 \leq \epsilon. \end{align}
I am interested in the following problem (problem 2): \begin{align} \text{minimize} &\hspace{1em} \| W x \|_1 \\ \text{subject to} & \hspace{1em} \| y - \Phi x \|_2 \leq \epsilon. \end{align}
If $W$ is invertible, then I can define $\tilde{x}=Wx$, which then becomes the following problem (in the form of problem 1): \begin{align} \text{minimize} & \hspace{1em} \| \tilde{x} \|_1 \\ \text{subject to} & \hspace{1em} \|y - \Phi W^{-1} \tilde{x} \|_2 \leq \epsilon. \end{align}
However, what happens when $W$ is not invertible. I can not find a way to convert the problem into the form of problem 1. Are there theorems of compressed sensing related to this situation? For example, if we know that the columns of $W$ are orthogonal, can that imply that $x$ will be recovered exactly? When?