Restricted Isometries for Partial Random Circulant Matrices (1010.1847v1)

Abstract: In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling.

• The paper establishes improved RIP bounds for partial random circulant matrices, showing that m ≳ s^(3/2) log^(3/2)(n) samples suffice for accurate recovery.
• It employs a variant of the Dudley inequality combined with Fourier analysis to rigorously control the complexity of Rademacher chaos processes.
• These insights enable more efficient compressed sensing strategies in practical applications such as radar imaging and optical signal processing.

Restricted Isometries for Partial Random Circulant Matrices

The paper "Restricted Isometries for Partial Random Circulant Matrices" by Holger Rauhut, Justin Romberg, and Joel A. Tropp offers a significant contribution to the analysis of efficient measurement matrices in compressed sensing. Specifically, the authors focus on partial random circulant matrices, which model measurement processes involving convolution with random pulses followed by non-random subsampling. This work improves upon existing analyses whose bounds do not exhibit optimal scaling.

Key Results and Methodology:

The central result asserts that the sth order restricted isometry constant of a partial random circulant matrix is small under certain conditions. Specifically, the paper establishes the bound $m \geq C\delta^{-1}s^{3/2}\log^{3/2}(n)$, where $m$ is the number of samples needed, $n$ is the length of the pulse, and $C$ is a constant. This bound is an improvement over previous estimates that exhibited quadratic scaling in $s$.

The authors leverage a variant of the classical Dudley inequality to perform their analysis. This approach involves bounding the expectation of the supremum of a Rademacher chaos process using entropy integrals related to covering numbers of particular metric spaces. By applying elementary results from Fourier analysis, they efficiently deal with the combinatorial complexity inherent in these processes.

Implications and Application:

Theoretical implications of this work include enhanced understanding of structured measurement matrices beyond Gaussian and Rademacher matrices, which are typically used due to their optimal RIP properties. The results suggest pathways for achieving efficient sample recovery in systems constrained by specific structural considerations.

From a practical standpoint, the paper's results could impact applications such as radar imaging and signal processing. For instance, in radar applications, the reduction in sampling rates can lead to the use of cheaper and more accurate ADCs without loss of resolution, given that the signal of interest is sparse. Additionally, the framework supports signal recovery in optical systems employing coded apertures to expand field-of-view, enabling high-resolution reconstruction from limited sensor data.

Future Directions:

Despite the advancements made in this paper, the authors acknowledge the gap between their results and the theoretically optimal scaling behavior desired in compressive sensing. Further research may focus on overcoming the identified bottlenecks, such as the sharpness of the Dudley-type inequality used. Exploration of other types of generating sequences and application of similar analytical techniques to more varied matrix structures could also prove fruitful. Moreover, extending these results to non-ideal and noisy measurement conditions remains an open area for investigation, especially in applications where robustness is as critical as accuracy.

Overall, this paper presents a detailed and considered advancement in the understanding of restricted isometries for structured random matrices, reinforcing the utility of compressive sensing in practical, technologically relevant scenarios.