Information theoretic bounds for Compressed Sensing (0804.3439v5)

Abstract: In this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections. We consider two settings: output noise models where the noise enters after the projection and input noise models where the noise enters before the projection. We consider two types of distortion for reconstruction: support errors and mean-squared errors. Our goal is to relate the number of measurements, $m$, and $\snr$, to signal sparsity, $k$, distortion level, $d$, and signal dimension, $n$. We consider support errors in a worst-case setting. We employ different variations of Fano's inequality to derive necessary conditions on the number of measurements and $\snr$ required for exact reconstruction. To derive sufficient conditions we develop new insights on max-likelihood analysis based on a novel superposition property. In particular this property implies that small support errors are the dominant error events. Consequently, our ML analysis does not suffer the conservatism of the union bound and leads to a tighter analysis of max-likelihood. These results provide order-wise tight bounds. For output noise models we show that asymptotically an $\snr$ of $\Theta(\log(n))$ together with $\Theta(k \log(n/k))$ measurements is necessary and sufficient for exact support recovery. Furthermore, if a small fraction of support errors can be tolerated, a constant $\snr$ turns out to be sufficient in the linear sparsity regime. In contrast for input noise models we show that support recovery fails if the number of measurements scales as $o(n\log(n)/SNR)$ implying poor compression performance for such cases. We also consider Bayesian set-up and characterize tradeoffs between mean-squared distortion and the number of measurements using rate-distortion theory.

• The paper analyzes compressed sensing using information-theoretic methods to derive fundamental bounds on measurements needed for signal reconstruction under output and input noise models.
• For the output noise model, exact signal recovery requires signal-to-noise ratio scaling as (log(n)) and measurements as (k log(n/k)).
• The input noise model shows poor compression performance, while a Bayesian approach leveraging rate-distortion theory can enable linear scaling of measurements with constant SNR.

Information Theoretic Bounds for Compressed Sensing

The paper "Information Theoretic Bounds for Compressed Sensing" by Shuchin Aeron, Venkatesh Saligrama, and Manqi Zhao proposes a comprehensive framework for analyzing the compressed sensing problem using information-theoretic approaches. This work is focused on deriving fundamental bounds on the number of measurements required for the exact and approximate reconstruction of sparse signals subject to different noise models, namely output and input noise models. The authors employ variations of Fano's inequality and novel maximum likelihood (ML) analyses to establish these bounds.

Key Contributions

1. Output Noise Model Bounds:
   • The authors demonstrate that for the output noise model, reconstruction of sparse signal support requires the signal-to-noise ratio (SNR) to scale as $\Theta(\log(n))$. Additionally, the number of measurements needed is $\Theta(k \log(n/k))$ for exact recovery.
   • For approximate support recovery, tolerating small support errors allows for achieving desired reconstruction performance with constant SNR in linear sparsity regimes.
2. Input Noise Model Results:
   • Contrary to the output noise model, the input noise model incurs a compression penalty where the number of measurements must scale as $o(n \log(n)/\text{SNR})$. This indicates poor compression performance under these conditions, as observed in sensor network applications, where noise enters before signal compression.
3. Bayesian Setup:
   • A Bayesian approach is employed to reduce the conservative requirements of the worst-case setup. This involves novel extensions to Fano's inequality capable of handling continuous domains and arbitrary distortions, harnessing the power of rate-distortion theory.
   • With constant SNR, the required number of measurements can scale linearly with the rate-distortion function of the sparse phenomenon, offering promising trade-offs between distortion level and measurement number.

Implications and Future Work

The results provided in this paper have significant implications for the theoretical understanding and practical application of compressed sensing. Firstly, understanding the fundamental bounds informs the design of more efficient sensing systems that are tailored to specific noise conditions. Secondly, the trade-offs highlighted in Bayesian setups suggest avenues for future research into probabilistic signal models that may allow for compressive sensing with fewer measurements.

This paper pushes the boundaries of compressed sensing theory by providing a rigorous, analytically grounded understanding of the interplay between SNR, sparsity, distortion, and measurements. Going forward, improvements in compression performance under input noise models and further exploration of Bayesian signal models could lead to enhanced sensing capabilities in diverse AI applications. Additionally, leveraging these bounds in real-world implementations could optimize the design of sensing matrices for tasks in image processing, medical diagnostics, and network sensor fusion systems.

Overall, the authors present a robust framework that not only delineates the theoretical limits of compressed sensing but also lays down foundational principles that could be adapted to emerging technological paradigms in artificial intelligence and data science.