  Level-set optimization formulations with data-driven constraints minimize a
regularization functional subject to matching observations to a given error
level. These formulations are widely used, particularly for matrix completion
and sparsity promotion in data interpolation and denoising. The misfit level is
typically measured in the l2 norm, or other smooth metrics. In this paper, we
present a new flexible algorithmic framework that targets nonsmooth level-set
constraints, including L1, Linf, and even L0 norms. These constraints give
greater flexibility for modeling deviations in observation and denoising, and
have significant impact on the solution. Measuring error in the L1 and L0 norms
makes the result more robust to large outliers, while matching many
observations exactly. We demonstrate the approach for basis pursuit denoise
(BPDN) problems as well as for extensions of BPDN to matrix factorization, with
applications to interpolation and denoising of 5D seismic data. The new methods
are particularly promising for seismic applications, where the amplitude in the
data varies significantly, and measurement noise in low-amplitude regions can
wreak havoc for standard Gaussian error models.
