Information-Theoretically Optimal Compressed Sensing via Spatial Coupling and Approximate Message Passing (1112.0708v2)

Abstract: We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala et al. \cite{KrzakalaEtAl}, message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate $\delta$ exceeds the (upper) R\'enyi information dimension of the signal, $\uRenyi(p_X)$. More precisely, for a sequence of signals of diverging dimension $n$ whose empirical distribution converges to $p_X$, reconstruction is with high probability successful from $\uRenyi(p_X)\, n+o(n)$ measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension $n$ and $k(n)$ non-zero entries, this implies reconstruction from $k(n)+o(n)$ measurements. For `discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from $o(n)$ measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal $p_X$.

• The paper establishes information-theoretically optimal compressed sensing by leveraging spatially coupled sensing matrices and an AMP algorithm.
• It introduces state evolution analysis to track noise and iteratively predict signal recovery, particularly for sparse signals.
• The approach significantly reduces required measurements relative to sparsity, promising robust applications in efficient data acquisition and signal processing.

Examination of Optimal Compressed Sensing through Spatial Coupling and Approximate Message Passing

The paper "Information-Theoretically Optimal Compressed Sensing via Spatial Coupling and Approximate Message Passing," authored by David L. Donoho, Adel Javanmard, and Andrea Montanari, explores advancing compressed sensing methodologies by leveraging spatial coupling and approximate message passing (AMP). This work builds upon previous findings in coding theory and compressed sensing to offer an efficient reconstruction algorithm that approaches the theoretical limits set by information theory.

Core Contribution and Methodology

The objective of the paper is to address the compressed sensing reconstruction problem, particularly for band-diagonal sensing matrices inspired by spatial coupling—a concept initially explored in error-correcting codes. The researchers employ an approximate message passing algorithm, analyzed through the state evolution method, to demonstrate that signal reconstruction is viable when the undersampling rate exceeds the Renyi information dimension of the signal.

Key to this approach is the realization that employing spatially coupled sensing matrices allows effective signal recovery from fewer measurements compared to traditional random matrices. Notably, for sparse signals, the framework enables reconstruction with a number of measurements almost equivalent to the number of non-zero entries. This is a significant departure from existing methods, which require more measurements reflecting the sparsity level.

The findings stand out due to the robustness against noise and the general applicability beyond random signals, contingent upon knowledge of the empirical distribution. The authors rigorously establish these results, underscoring the efficacy of the AMP algorithm in tracking the state of noise and prediction iteratively, exploiting local statistics furnished by spatial coupling.

Evaluation and Theoretical Insights

The research is encapsulated in formal theorems and lemmas that together assert robust and near-optimal compressed sensing capabilities under specific conditions related to signal undersampling. The implications reach beyond immediate practical use, highlighting a shift in the fundamental structure facilitating signal reconstruction. The focus is redirected from sparsity alone to a broader notion of structure as quantified by information dimension.

Consequently, these findings hold substantial implications for fields relying on signal processing and data acquisition. The paper posits its methodology as a potential game-changer for applications demanding high efficiency in scenarios constrained by limited data acquisition capacity.

Future Directions

Given the rigorous theoretical framework and the promising results, several future research paths emerge. Potential adaptations include the extension of these principles to more diverse signal models or environments with differing noise characteristics. Moreover, developing more computationally viable versions of the AMP algorithm that do not strictly rely on precise prior distributions could expand its applicability.

The intersections of spatially coupled designs in compressed sensing with other domains, such as machine learning or data-driven decision systems, may also offer intriguing avenues. The integration of such algorithms with real-world systems, potentially leading to dynamic, near-real-time data processing solutions, could significantly enhance various technological capabilities.

In conclusion, this paper provides a robust theoretical canon and introduces a promising approach in compressed sensing. The use of spatial coupling augmented with message passing considerably narrows the gap to theoretical performance limits, heralding advancements across numerous applications relying on effective and efficient data reconstruction.