  Inspired by significant real-life applications, in particular, sparse phase
retrieval and sparse pulsation frequency detection in Asteroseismology, we
investigate a general framework for compressed sensing, where the measurements
are quasi-linear. We formulate natural generalizations of the well-known
Restricted Isometry Property (RIP) towards nonlinear measurements, which allow
us to prove both unique identifiability of sparse signals as well as the
convergence of recovery algorithms to compute them efficiently. We show that
for certain randomized quasi-linear measurements, including Lipschitz
perturbations of classical RIP matrices and phase retrieval from random
projections, the proposed restricted isometry properties hold with high
probability. We analyze a generalized Orthogonal Least Squares (OLS) under the
assumption that magnitudes of signal entries to be recovered decay fast. Greed
is good again, as we show that this algorithm performs efficiently in phase
retrieval and asteroseismology. For situations where the decay assumption on
the signal does not necessarily hold, we propose two alternative algorithms,
which are natural generalizations of the well-known iterative hard and
soft-thresholding. While these algorithms are rarely successful for the
mentioned applications, we show their strong recovery guarantees for
quasi-linear measurements which are Lipschitz perturbations of RIP matrices.
