Ideas on Matrix Factorization / Transformations for $ {L}_{1} $ Minimization I am starting with a typical $\ell_1$ basis pursuit problem:

$$
\min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon,
$$

where $\mathbf{R}\in\mathbb{C}^{M \times P}$, and $\mathbf{E}$ is an $M' \times M$ row selection matrix that randomly chooses $M'&lt;P$ rows from $\mathbf{R}$. It is also known that $\operatorname{rank}(\mathbf{ER})=M'$, so that the constraint system is full-rank but under-determined.

Since both $M$ and $P$ are on the order of $10^7$, the above problem is impractical because of memory and computational limitations. An algorithm exists that will let me efficiently compute $\mathbf{R^{\mathrm{H}}}\mathbf{E^{\mathrm{H}}}\mathbf{y}$, so I would like to see if I can combine this with a sparse approximation (described below) to solve the following equivalent problem:

$$
\min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{R^{\mathrm{H}}}\mathbf{E^{\mathrm{H}}}\mathbf{ERx} - \mathbf{R^{\mathrm{H}}}\mathbf{E^{\mathrm{H}}}\mathbf{y} \Vert_2 \leq \epsilon,
$$

Since $\mathbf{E}$ is a row-selection matrix, the product $\mathbf{E^{\mathrm{H}}}\mathbf{E}$ is a diagonal matrix with a $1$ at element $(i,i)$ if row $i$ was selected and $0$ otherwise. Furthermore, given the nature of $\mathbf{R}$, even though the product $\mathbf{R^{\mathrm{H}}}\mathbf{E^{\mathrm{H}}}\mathbf{ER}$ is rather large at $P \times P$, it is also approximately sparse with only a relatively small number of terms with significant modulus per row/column. This makes me think that the above form is doable since I could compute just the sparse approximation and avoid the unrealistic memory and computation requirements.

However, $\mathbf{R^{\mathrm{H}}}\mathbf{E^{\mathrm{H}}}\mathbf{ER}$ has size $P \times P$ but is only of rank $M'$ and this is causing problems with the  basis pursuit algorithms as they expect the rows of the constraint matrix to be
linearly independent. Hence, my solicitation for input.

Are there any factorizations and/or transformations that could turn the constraints of the second form of the problem into under-determined yet full-rank system? Please do keep in mind the large sizes of these matrices. Any other suggestions/comments are also welcome. We can assume that $\mathbf{R}$ has rank $M$ or $P$, as there is freedom in its design.
Since $ \epsilon $ is a parameter you need to set why not trade it with another parameter you need to set to create an easily solvable problem (Relaxation of the Problem)?

You can transform the problem into the following form ($ {L}_{1} $ Regularized Least Squares):

$$ \arg \min_{x} \frac{1}{2} \left\| A x - z \right\|^{2} + \lambda \left\| x \right\|_{1} $$  

There are many solvers for that even for large sparse matrices.
For any $ \epsilon $ from the original problem there is $ \lambda $ such that the solutions are the same.
Since $ \epsilon $ it to be arbitrarily set in the original problem you can trade it for $ \lambda $.

This can be solved with many solvers easily without the need for Matrix Inversion.