  This work deals with the problem of distributed data acquisition under
non-linear communication constraints. More specifically, we consider a model
setup where $M$ distributed nodes take individual measurements of an unknown
structured source vector $x_0 \in \mathbb{R}^n$, communicating their readings
simultaneously to a central receiver. Since this procedure involves collisions
and is usually imperfect, the receiver measures a superposition of non-linearly
distorted signals. In a first step, we will show that an $s$-sparse vector
$x_0$ can be successfully recovered from $O(s \cdot\log(2n/s))$ of such
superimposed measurements, using a traditional Lasso estimator that does not
rely on any knowledge about the non-linear corruptions. This direct method
however fails to work for several "uncalibrated" system configurations. These
blind reconstruction tasks can be easily handled with the
$\ell^{1,2}$-Group-Lasso, but coming along with an increased sampling rate of
$O(s\cdot \max\{M, \log(2n/s) \})$ observations - in fact, the purpose of this
lifting strategy is to extend a certain class of bilinear inverse problems to
non-linear acquisition. Our two algorithmic approaches are a special instance
of a more abstract framework which includes sub-Gaussian measurement designs as
well as general (convex) structural constraints. These results are of
independent interest for various recovery and learning tasks, as they apply to
arbitrary non-linear observation models. Finally, to illustrate the practical
scope of our theoretical findings, an application to wireless sensor networks
is discussed, which actually serves as the prototypical example of our
methodology.
