  Sparse coding and dictionary learning are popular techniques for linear
inverse problems such as denoising or inpainting. However in many cases, the
measurement process is nonlinear, for example for clipped, quantized or 1-bit
measurements. These problems have often been addressed by solving constrained
sparse coding problems, which can be difficult to solve, and assuming that the
sparsifying dictionary is known and fixed. Here we propose a simple and unified
framework to deal with nonlinear measurements. We propose a cost function that
minimizes the distance to a convex feasibility set, which models our knowledge
about the nonlinear measurement. This provides an unconstrained, convex, and
differentiable cost function that is simple to optimize, and generalizes the
linear least squares cost commonly used in sparse coding. We then propose
proximal based sparse coding and dictionary learning algorithms, that are able
to learn directly from nonlinearly corrupted signals. We show how the proposed
framework and algorithms can be applied to clipped, quantized and 1-bit data.
