**Abstract:** The restricted isometry property (RIP) has been an integral tool in the analysis of various inverse problems with sparsity models in signal processing and statistics. We propose generalized notions of sparsity and provide a unified framework for the RIP of structured random measurements given by isotropic group actions. Our results extend the RIP for partial Fourier measurements by Rudelson and Vershynin to a much broader context and identify a sufficient number of group structured measurements for the RIP on generalized sparsity models. We illustrate the main results on an infinite dimensional example, where the sparsity represented by a smoothness condition approximates the total variation. We also discuss fast dimensionality reduction on generalized sparsity models. In generalizing sparsity models, the parameter accounting for the level of sparsity becomes no longer sub-additive. Therefore, the RIP does not preserve distances between sparse vectors. We show a weaker version with additive distortion, which is similar to analogous property arising in the 1-bit compressed sensing problem. This is a joint work with Marius Junge.