Model-Based Compressive Sensing (0808.3572v5)

Abstract: Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for the acquisition of sparse or compressible signals that can be well approximated by just K << N elements from an N-dimensional basis. Instead of taking periodic samples, CS measures inner products with M < N random vectors and then recovers the signal via a sparsity-seeking optimization or greedy algorithm. Standard CS dictates that robust signal recovery is possible from M = O(K log(N/K)) measurements. It is possible to substantially decrease M without sacrificing robustness by leveraging more realistic signal models that go beyond simple sparsity and compressibility by including structural dependencies between the values and locations of the signal coefficients. This paper introduces a model-based CS theory that parallels the conventional theory and provides concrete guidelines on how to create model-based recovery algorithms with provable performance guarantees. A highlight is the introduction of a new class of structured compressible signals along with a new sufficient condition for robust structured compressible signal recovery that we dub the restricted amplification property, which is the natural counterpart to the restricted isometry property of conventional CS. Two examples integrate two relevant signal models - wavelet trees and block sparsity - into two state-of-the-art CS recovery algorithms and prove that they offer robust recovery from just M=O(K) measurements. Extensive numerical simulations demonstrate the validity and applicability of our new theory and algorithms.

• The paper introduces structured sparsity models that lower measurement needs compared to traditional compressive sensing.
• It integrates these models into algorithms like CoSaMP and IHT for enhanced recovery performance in wavelet and block-sparse applications.
• Empirical results demonstrate that the framework reduces sample complexity from O(K log(N/K)) to O(K) or O(JK), improving efficiency in practical settings.

Model-Based Compressive Sensing

The paper "Model-Based Compressive Sensing" by Baraniuk et al. presents an advanced theoretical framework for Compressive Sensing (CS) that exploits structural signal models. Building upon conventional CS, where signals are acquired via measurements using inner products with random vectors, the paper introduces a novel approach that leverages more sophisticated signal models to reduce measurement requirements while maintaining robust recovery properties.

Overview

Compressive Sensing traditionally relies on the sparsity and compressibility of signals, enabling recovery from fewer measurements than dictated by Shannon/Nyquist sampling. Standard CS necessitates approximately $M = O(K \log(N/K))$ measurements for robust recovery of $K$-sparse signals in an $N$-dimensional space. This paper posits that by integrating structured sparsity models—beyond mere sparsity—substantial reductions in measurements $M$ can be achieved without sacrificing robustness.

Core Contributions

The fundamental advancements discussed in the paper include:

1. Introduction of Structural Models: The paper extends traditional CS by incorporating structured sparsity models, which include dependencies between the values and locations of signal coefficients. These models offer a more accurate representation of realistic signal behavior compared to simple sparsity.
2. Model-Based Restricted Isometry Property (RIP): A new theoretical framework generalizes the RIP to account for structured sparsity models, defining the restricted amplification property (RAmP) for robust recovery of structured compressible signals. The RAmP facilitates the derivation of conditions under which the measurement matrix retains essential signal information.
3. Algorithmic Integration: The authors integrate these structural models into existing CS algorithms, specifically adapting CoSaMP and Iterative Hard Thresholding (IHT). The modification involves replacing the standard $K$-term approximation with a structured sparse approximation step, thereby embedding model-based recovery directly into the iterative process.

Numerical Results

The paper substantiates the theoretical claims through extensive numerical simulations, demonstrating the efficacy of model-based recovery algorithms:

• Tree-Based Models: For wavelet-sparse signals, the paper uses models that capture the tree structure of significant wavelet coefficients. Notably, the tree-based CoSaMP algorithm requires only $M = O(K)$ measurements, substantiated by empirical results that highlight superior performance compared to conventional CoSaMP, particularly in low-measurement regimes.

Moreover, the experiments show that model-based recovery retains robustness in noisy environments, with performance comparable to standard methods but with reduced measurement requirements.

• Block-Sparse Signals: Addressing block-sparse signals and signal ensembles with shared support, the block-based CoSaMP algorithm similarly reduces measurement overhead to $M = O(JK)$, where $JK$ represents the total sparsity. Empirical validation confirms the theoretical improvements, emphasizing the practical utility of the block-sparse models.

Theoretical and Practical Implications

The theoretical advancements discussed in the paper have profound implications:

• Reduction in Measurement Requirements: By leveraging structured sparsity models, significant reductions in the number of measurements required for stable and robust signal recovery are achieved. This advancement is particularly critical for applications where sampling costs or data acquisition rates are prohibitionally high.
• Extension to Various Signal Models: The framework set forth by the authors enables the extension of CS to various complex signal models, thereby broadening the applicability of CS in real-world scenarios. This includes high-dimensional polytopes and nonlinear manifolds, potentially revolutionizing fields such as medical imaging, remote sensing, and signal processing.
• Algorithmic Modifications: The direct integration of structured sparsity models into iterative recovery algorithms opens new avenues for efficiently solving larger and more complex compressive sensing problems. This could lead to new algorithmic developments harnessing the power of structured models.

Future Directions

The proposed model-based CS framework sparks several avenues for future research:

• Development of New Structured Models: Identifying and formalizing additional structured models that capture the intrinsic properties of diverse signal classes can further optimize CS performance across various domains.
• Optimization-Based Methods: Integrating structured sparsity into convex optimization-based recovery methods (such as basis pursuit) could offer alternative pathways to enhance the robustness and efficiency of CS.
• Real-Time Applications: Extending model-based CS frameworks to real-time applications necessitates research into computational optimizations and hardware implementations, ensuring that theoretical advances translate into practical benefits.

In conclusion, "Model-Based Compressive Sensing" marks a significant progression in the domain of signal processing, providing a robust and efficient framework that leverages structured sparsity models. This work not only enhances the theoretical foundations of CS but also paves the way for practical advancements that can cater to the increasing demands of high-fidelity signal acquisition and processing.