  We study the estimation capacity of the generalized Lasso, i.e., least
squares minimization combined with a (convex) structural constraint. While
Lasso-type estimators were originally designed for noisy linear regression
problems, it has recently turned out that they are in fact robust against
various types of model uncertainties and misspecifications, most notably,
non-linearly distorted observation models. This work provides more theoretical
evidence for this somewhat astonishing phenomenon. At the heart of our analysis
stands the mismatch principle, which is a simple recipe to establish
theoretical error bounds for the generalized Lasso. The associated estimation
guarantees are of independent interest and are formulated in a fairly general
setup, permitting arbitrary sub-Gaussian data, possibly with strongly
correlated feature designs; in particular, we do not assume a specific
observation model which connects the input and output variables. Although the
mismatch principle is conceived based on ideas from statistical learning
theory, its actual application area are (high-dimensional) estimation tasks for
semi-parametric models. In this context, the benefits of the mismatch principle
are demonstrated for a variety of popular problem classes, such as single-index
models, generalized linear models, and variable selection. Apart from that, our
findings are also relevant to recent advances in quantized and distributed
compressed sensing.
