Basis Pursuit Denoise With Nonsmooth Constraints
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 <inline-formula><tex-math notation="LaTeX">$\ell _2$</tex-math></inline-formula> norm, or other smooth metrics. In this paper, we present a new flexible algorithmic framework that targets <italic>nonsmooth</italic> level-set constraints, including <inline-formula><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$\ell _\infty$</tex-math></inline-formula>, and even <inline-formula><tex-math notation="LaTeX">$\ell _0$</tex-math></inline-formula> norms. These constraints give greater flexibility for modeling deviations in observation and denoising, and have significant impact on the solution. Measuring error in the <inline-formula><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\ell _0$</tex-math></inline-formula> 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 4D 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.