Before revision:   Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising (BPDN) . If the measurement matrix has the robust null space property with respect to the \ell_2-norm, BPDN obeys stable and robust recovery guarantees.
After revision:  The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising (BPDN) . If the measurement matrix has the robust null space property with respect to the \ell_2-norm, BPDN obeys stable and robust recovery guarantees.