  In the context of sketching for compressive mixture modeling, we revisit
existing proofs of the Restricted Isometry Property of sketching operators with
respect to certain mixtures models. After examining the shortcomings of
existing guarantees, we propose an alternative analysis that circumvents the
need to assume importance sampling when drawing random Fourier features to
build random sketching operators. Our analysis is based on new deterministic
bounds on the restricted isometry constant that depend solely on the set of
frequencies used to define the sketching operator; then we leverage these
bounds to establish concentration inequalities for random sketching operators
that lead to the desired RIP guarantees. Our analysis also opens the door to
theoretical guarantees for structured sketching with frequencies associated to
fast random linear operators.
