On random and deterministic compressed sensing and the Restricted Isometry Property in levels

Compressed sensing (CS) is one of the great successes of computational mathematics in the past decade. There are a collection of tools which aim to mathematically describe compressed sensing when the sampling pattern is taken in a random or deterministic way. Unfortunately, there are many practical applications where the well studied concepts of uniform recovery and the Restricted Isometry Property (RIP) can be shown to be insufficient explanations for the success of compressed sensing. This occurs both when the sampling pattern is taken using a deterministic or a non-deterministic method. We shall study this phenomenon and explain why the RIP is absent, and then propose an adaptation which we term `the RIP in levels' which aims to solve the issues surrounding the RIP. The paper ends by conjecturing that the RIP in levels could provide a collection of results for deterministic sampling patterns.