A probabilistic and RIPless theory of compressed sensing

Abstract: This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all models - e.g. Gaussian, frequency measurements - discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) - they make use of a much weaker notion - or a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.

• The paper introduces a RIPless probabilistic framework that recovers approximately sparse signals via l1 minimization from on the order of s log n measurements.
• It leverages isotropy and incoherence properties to ensure stable recovery even when measurements are noisy.
• The methodology employs advanced probabilistic tools and a golfing scheme, opening new avenues for practical applications in imaging and signal processing.

A Probabilistic and RIPless Theory of Compressed Sensing

The paper by Emmanuel J. Candès and Yaniv Plan introduces a novel framework for compressive sensing that deviates from conventional approaches by not relying on the Restricted Isometry Property (RIP). The authors propose a probabilistic model where sensing vectors are independently sampled from a specific probability distribution, allowing for signal recovery from fewer measurements, even when these measurements are noisy. This work provides a more flexible foundation for developing new measurement strategies and demonstrates the feasibility of a simpler recovery theory broadly applicable to existing models such as Gaussian and Fourier.

Key Contributions

The authors address fundamental challenges in compressive sensing regarding the recovery of approximately sparse signals from limited and noisy measurements. The framework they introduce revolves around two main properties of the sensing distributions:

1. Isotropy Property: This ensures that the sensing vectors, when sampled, maintain unit variance and are uncorrelated. The conditional expectation and variance framework underpin much of compressed sensing, and the authors extend it by allowing for near-isotropy.
2. Incoherence Property: This measures the degree to which the vectors are uniformly distributed. A low coherence implies fewer measurements are needed to reconstruct the signals accurately.

One of the paper's significant results is proving that approximately sparse signals can be recovered using $\ell_1$ minimization from a minimal number of measurements on the order of $s \log n$, where $s$ is the sparsity level and $n$ the ambient dimension, without relying on RIP.

Methodological Framework

The authors utilize advanced probabilistic tools and matrix concentration inequalities to establish the main theoretical results. By introducing the concept of a "golfing scheme" and leveraging techniques from majorizing measures, they are able to craft stable guarantees for the recovery of sparse signals in the absence of RIP.

Implications and Future Directions

The introduction of a RIPless framework allows compressive sensing to be applied more broadly, where RIP verification is computationally demanding or unenforceable. This research has immediate practical implications in areas such as medical imaging, particularly MRI, where the random measurement model aligns well with existing sampling strategies. Furthermore, the probabilistic model enables the exploration of new sensing matrices beyond traditional paradigms.

Theoretically, this work may inspire further exploration of non-RIP frameworks, potentially reducing the complexity and increasing the efficiency of compressive sensing algorithms. The adaptability of this methodology to different distributions will likely invoke further research in expanding the types of distributions that can successfully employ compressive sensing strategies.

In summary, Candès and Plan have laid important groundwork that challenges traditional assumptions in compressive sensing, offering a new perspective that balances simplicity and generality. This approach opens up new avenues for both theoretical exploration and practical application across various domains where data acquisition and processing efficiency are critical.