  Quadratically-constrained basis pursuit has become a popular device in sparse
regularization; in particular, in the context of compressed sensing. However,
the majority of theoretical error estimates for this regularizer assume an a
priori bound on the noise level, which is usually lacking in practice. In this
paper, we develop stability and robustness estimates which remove this
assumption. First, we introduce an abstract framework and show that robust
instance optimality of any decoder in the noise-aware setting implies stability
and robustness in the noise-blind setting. This is based on certain sup-inf
constants referred to as quotients, strictly related to the quotient property
of compressed sensing. We then apply this theory to prove the robustness of
quadratically-constrained basis pursuit under unknown error in the cases of
random Gaussian matrices and of random matrices with heavy-tailed rows, such as
random sampling matrices from bounded orthonormal systems. We illustrate our
results in several cases of practical importance, including subsampled Fourier
measurements and recovery of sparse polynomial expansions.
