Near-Optimal Adaptive Compressed Sensing (1306.6239v2)

Abstract: This paper proposes a simple adaptive sensing and group testing algorithm for sparse signal recovery. The algorithm, termed Compressive Adaptive Sense and Search (CASS), is shown to be near-optimal in that it succeeds at the lowest possible signal-to-noise-ratio (SNR) levels, improving on previous work in adaptive compressed sensing. Like traditional compressed sensing based on random non-adaptive design matrices, the CASS algorithm requires only k log n measurements to recover a k-sparse signal of dimension n. However, CASS succeeds at SNR levels that are a factor log n less than required by standard compressed sensing. From the point of view of constructing and implementing the sensing operation as well as computing the reconstruction, the proposed algorithm is substantially less computationally intensive than standard compressed sensing. CASS is also demonstrated to perform considerably better in practice through simulation. To the best of our knowledge, this is the first demonstration of an adaptive compressed sensing algorithm with near-optimal theoretical guarantees and excellent practical performance. This paper also shows that methods like compressed sensing, group testing, and pooling have an advantage beyond simply reducing the number of measurements or tests -- adaptive versions of such methods can also improve detection and estimation performance when compared to non-adaptive direct (uncompressed) sensing.

• The paper presents Compressive Adaptive Sense and Search (CASS), a novel adaptive sensing algorithm for near-optimal sparse signal recovery.
• CASS achieves successful signal recovery at SNR levels a factor of log n lower than standard non-adaptive techniques, requiring approximately 2k log2 (n/k) measurements.
• The algorithm demonstrates better reconstruction quality in experiments and is highly efficient for high-dimensional data recovery in noisy or computationally constrained scenarios.

Near-Optimal Adaptive Compressed Sensing

This paper presents a novel adaptive sensing algorithm known as Compressive Adaptive Sense and Search (CASS) for sparse signal recovery. The proposed algorithm demonstrates theoretical near-optimality in terms of signal-to-noise ratio (SNR) requirements, accomplishing successful recovery at SNR levels that are a factor of $\log n$ lower than those needed by standard non-adaptive compressed sensing techniques. This breakthrough is particularly significant for applications involving high-dimensional data.

Technical Contributions

The central contribution of this research is the development of a simple adaptive procedure for signal recovery that asserts optimal theoretical guarantees and strong empirical performance. The procedure effectively isolates non-zero components of sparse signals by leveraging adaptively designed sensing vectors. A key innovation in the CASS algorithm is the strategic allocation of sensing energy across multiple steps of adaptive measurement. This allocation reduces computational overhead compared to traditional compressed sensing techniques, which often necessitate dense sensing matrices and elaborate reconstruction algorithms.

Theoretical Results

The paper claims that the CASS algorithm ensures exact support recovery of non-negative signals under certain conditions. Specifically, given a $k$-sparse signal of length $n$, the algorithm requires approximately $m = 2k \log_2 (n/k)$ measurements and guarantees correct identification of signal support if the minimum amplitude of non-zero entries satisfies:

$$x_{\min} \geq \sqrt{20 \frac{n}{M} \left( \log k + \log \left( \frac{8}{\delta}\right) \right)}$$

Here, $M$ denotes the total sensing energy available, and $\delta$ represents the probability of error. The guarantees extend to the recovery of signals with both positive and negative entries, provided the sparsity pattern is chosen uniformly at random and allowing a scalable mechanism to control the expected symmetric set difference in support recovery.

Practical Implications

The results have profound implications for compressed sensing applications, particularly in areas where signal acquisition is limited by noise, or where the dimensionality of data imposes constraints on computational resources. Signals such as images, which are often sparse in some transform domain (e.g., wavelets), can be analyzed more efficiently with CASS, as demonstrated through numerical experiments on image datasets. In practice, CASS consistently achieved better reconstruction quality than standard compressed sensing using LASSO recovery, even when only modest levels of measurement noise were present.

Future Directions

Potential avenues for future research include optimizing the constant factor in the SNR conditions required for the success of CASS, thus improving its practical applicability across different signal types and domains. Furthermore, adapting the CASS framework to align with signals sparse in overcomplete dictionaries or extending its use to domains outside traditional signal processing, such as network science and bioinformatics, could drive innovative applications.

In conclusion, the CASS algorithm provides a significant advancement in the field of compressed sensing, enabling effective signal recovery in scenarios constrained by low SNR and offering substantial benefits over conventional non-adaptive approaches. Its interplay of theoretical insights and empirical results ensures it stands out as a versatile tool for modern signal processing challenges.