Blind Compressed Sensing (1002.2586v2)

Abstract: The fundamental principle underlying compressed sensing is that a signal, which is sparse under some basis representation, can be recovered from a small number of linear measurements. However, prior knowledge of the sparsity basis is essential for the recovery process. This work introduces the concept of blind compressed sensing, which avoids the need to know the sparsity basis in both the sampling and the recovery process. We suggest three possible constraints on the sparsity basis that can be added to the problem in order to make its solution unique. For each constraint we prove conditions for uniqueness, and suggest a simple method to retrieve the solution. Under the uniqueness conditions, and as long as the signals are sparse enough, we demonstrate through simulations that without knowing the sparsity basis our methods can achieve results similar to those of standard compressed sensing, which relay on prior knowledge of the sparsity basis. This offers a general sampling and reconstruction system that fits all sparse signals, regardless of the sparsity basis, under the conditions and constraints presented in this work.

• The paper introduces Blind Compressed Sensing, eliminating the requirement for known sparsity bases in signal recovery.
• It establishes uniqueness conditions by imposing constraints like finite basis sets, sparse representations, and orthogonal block-diagonal structures.
• The authors develop and validate algorithms (F-BCS, sparse K-SVD, OBD-BCS) that demonstrate competitive performance in blind signal reconstruction.

An Overview of Blind Compressed Sensing

The paper entitled "Blind Compressed Sensing" by Sivan Gleichman and Yonina C. Eldar introduces an innovative framework within the field of compressed sensing (CS), wherein the dependence on prior knowledge of the sparsity basis is eliminated. The core problem addressed is the recovery of a sparse signal from a limited number of linear measurements, using neither the sparsity basis for sampling nor recovery processes. This approach, termed Blind Compressed Sensing (BCS), merges concepts from both compressed sensing and dictionary learning (DL), exploring conditions under which such blind recovery is achievable.

Key Contributions and Results

1. Conceptual Framework: The authors challenge the traditional CS paradigm that relies on a known sparsity basis, proposing BCS, where recovery is possible under unknown sparsity conditions. The novelty lies in handling signals that are sparse under an unknown basis, while traditional CS requires this knowledge for successful signal recovery.
2. Uniqueness Conditions: To address the ill-posed nature of the blind recovery problem, the paper introduces specific constraints on the sparsity basis:
   • The sparsity basis is contained within a known finite set of bases.
   • The basis itself is sparse under a known dictionary.
   • The basis is orthogonal with a block-diagonal structure.

Each constraint is methodically analyzed, and conditions for uniqueness of the solution to BCS are rigorously derived, ensuring that possible ambiguities in signal reconstruction are mitigated.

1. Algorithmic Solutions: A significant aspect of the paper is the development of concrete algorithms tailored to each proposed constraint framework:
   • F-BCS Method: Engages traditional CS algorithms iteratively over a finite set of potential bases.
   • Sparse Basis Method: Utilizes a direct approach and sparse K-SVD to exploit basis sparsity.
   • OBD-BCS (Orthogonal Block Diagonal BCS): Leverages a multi-step algorithm to update the basis and sparse coefficients, effectively handling the block-diagonal structure to ensure reliable recovery.
2. Theoretical and Empirical Insights: Through comprehensive mathematical treatment and simulations, the authors demonstrate that under suitable sparsity and structural conditions, the BCS framework can achieve performance akin to traditional CS that benefits from a priori known bases. Notwithstanding some loss in performance due to the blind nature of the setup, the rampant versatility of BCS in unknown environments is unmistakably established.

Implications and Future Outlook

The development of BCS enhances the flexibility and applicability of compressed sensing by removing the restrictive requirement of known sparsity bases. This has pertinent implications for fields that deal with dynamic or complex signal scenarios where basis knowledge is impractical or infeasible, such as wireless communications, sensor networks, and medical imaging. Furthermore, the theoretical groundwork laid by this paper opens several avenues for future exploration:

• Algorithmic Efficiency: There is potential to improve computational aspects of the proposed algorithms, particularly in high-dimensional settings or when confronted with large datasets.
• Broader Class of Bases: Extending the framework to accommodate a broader class of random or structured bases, perhaps with adaptive algorithms, could yield further enhancements.
• Robustness and Noise Resilience: Investigating the resilience of BCS methods under various noise models and sparsity distributions remains an intriguing area for further research.

In conclusion, the paper makes a substantial contribution to the field of compressed sensing by introducing the BCS framework, proving its theoretical feasibility, and confirming its practical applicability through simulations. The work challenges existing paradigms and sets the stage for innovative approaches in data acquisition and signal processing, where traditional assumptions may no longer hold.