On the Absence of Uniform Recovery in Many Real-World Applications of Compressed Sensing and the Restricted Isometry Property and Nullspace Property in Levels

Abstract

The purpose of this paper is twofold. The first is to point out that the Restricted Isometry Property (RIP) does not hold in many applications where compressed sensing is successfully used. This includes fields like Magnetic Resonance Imaging (MRI), Computerized Tomography, Electron Microscopy, Radio Interferometry and Fluorescence Microscopy. We demonstrate that for natural compressed sensing matrices involving a level based reconstruction basis (e.g. wavelets), the number of measurements required to recover all s-sparse signals for reasonable s is excessive. In particular, uniform recovery of all s-sparse signals is quite unrealistic. This realisation shows that the RIP is insufficient for explaining the success of compressed sensing in various practical applications. The second purpose of the paper is to introduce a new framework based on a generalised RIP-like definition that fits the applications where compressed sensing is used. We show that the shortcomings that show that uniform recovery is unreasonable no longer apply if we instead ask for structured recovery that is uniform only within each of the levels. To examine this phenomenon, a new tool, termed the 'Restricted Isometry Property in Levels' is described and analysed. Furthermore, we show that with certain conditions on the Restricted Isometry Property in Levels, a form of uniform recovery within each level is possible. Finally, we conclude the paper by providing examples that demonstrate the optimality of the results obtained.We will thoroughly document the lack of RIP of order s in this paper, and explain why it does not hold for reasonable s. It is then natural to ask whether there might be an alternative to the RIP that may be more suitable for the actual real world CS applications. With this in mind, we shall introduce the RIP in levels which generalises the classical RIP and is much better suited for the actual applications where CS is used.

Breaking the Coherence Barrier: A New Theory for Compressed Sensing

“…There have also been several theoretical extensions. First, generalizations of the RIP for the setting of asymptotic sparsity, asymptotic incoherence and multilevel random subsampling have been introduced and analysed in [9,60,80]. These complement the results proved in this paper by establishing uniform recovery guarantees.…”

Correcting for unknown errors in sparse high-dimensional function approximation

“…We prove a robustness under unknown error for the WSR-LASSO decoder (11). Our theory suggests that the tuning parameter should be chosen directly proportional to the quantity K(s).…”

“…Then, the following holds with probability at least 1 − ε. For any f ∈ L 2 ν (D) ∩ L ∞ (D) expanded as in (3), the approximationf defined as in (13) computed using the WSR-LASSO decoder (11) with tuning parameter…”

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

“…Now that we have a good concept of sparsity in a level based reconstruction basis, it is time to define an analogue to the RIP. The natural adaptation, which we term the 'RIP in levels' (as described in [12]), is defined as follows:…”

“…It is worth noting that although Theorem 2 reduces to a standard result on the RIP in the case that l = 1, the requirement on δ s,M involves the number of levels l and a parameter η s,M . Further work in [12] has shown that this requirement cannot be avoided.…”