Recipes for Stable Linear Embeddings From Hilbert Spaces to $ {\mathbb {R}}^{m}$

We consider the problem of constructing a linear map from a Hilbert space H (possibly infinite dimensional) to ℝ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> that satisfies a restricted isometry property (RIP) on an arbitrary signal model, i.e., a subset of H. We present a generic framework that handles a large class of low-dimensional subsets but also unstructured and structured linear maps. We provide a simple recipe to prove that a random linear map satisfies a general RIP with high probability. We also describe a generic technique to construct linear maps that satisfy the RIP. Finally, we detail how to use our results in several examples, which allow us to recover and extend many known compressive sampling results.