Information-theoretic limits on sparse signal recovery: Dense versus sparse measurement matrices (0806.0604v1)

Abstract: We study the information-theoretic limits of exactly recovering the support of a sparse signal using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of observations $n$, the ambient signal dimension $p$, and the signal sparsity $k$ are all allowed to tend to infinity in a general manner. This paper makes two novel contributions. First, we provide sharper necessary conditions for exact support recovery using general (non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions, this result yields sharp characterizations of when the optimal decoder can recover a signal for various scalings of the sparsity $k$ and sample size $n$, including the important special case of linear sparsity ($k = \Theta(p)$) using a linear scaling of observations ($n = \Theta(p)$). Our second contribution is to prove necessary conditions on the number of observations $n$ required for asymptotically reliable recovery using a class of $\gamma$-sparsified measurement matrices, where the measurement sparsity $\gamma(n, p, k) \in (0,1]$ corresponds to the fraction of non-zero entries per row. Our analysis allows general scaling of the quadruplet $(n, p, k, \gamma)$, and reveals three different regimes, corresponding to whether measurement sparsity has no effect, a minor effect, or a dramatic effect on the information-theoretic limits of the subset recovery problem.

• The paper establishes sharp necessary conditions for exact support recovery using dense measurement matrices, including non-Gaussian ensembles.
• It derives precise conditions for recovery performance with γ-sparsified matrices, revealing distinct regimes based on measurement sparsity.
• The findings highlight trade-offs between statistical efficiency and computational cost, impacting applications in compressive sensing and signal denoising.

Information-Theoretic Limits on Sparse Signal Recovery: Dense Versus Sparse Measurement Matrices

This paper provides a thorough investigation into the information-theoretic limits of recovering the support of sparse signals using various noisy measurement matrices. The research is rooted in high-dimensional analysis, allowing the number of observations $n$, the ambient signal dimension $p$, and the signal sparsity $k$ to scale to infinity.

The two primary contributions of this work are:

1. Sharp Necessary Conditions for Dense Measurement Matrices: The paper presents tighter necessary conditions for exact support recovery using dense measurement matrices, including non-Gaussian ensembles. This extends known sufficient conditions from previous literature and allows for precise characterization of when optimal decoders can successfully recover sparse signals. Notably, the analysis covers scenarios of both linear sparsity, where $k = \Theta(p)$, and linear scaling of observations, $n = \Theta(p)$.
2. Conditions for Sparse Measurement Matrices: An intriguing aspect of the paper is the exploration of $$-sparsified measurement matrices, which are defined by the fraction$$\gamma$ of non-zero entries per row. This paper reveals three distinct regimes regarding the effect of measurement sparsity on recovery performance, providing necessary conditions for asymptotically reliable signal support recovery. These conditions illustrate whether measurement sparsity minimally, moderately, or significantly impacts the ability to recover signals from noisy measurements.

Implications and Future Directions

The results offer critical insights into the trade-offs between measurement sparsity and statistical efficiency. Dense matrices, such as the standard Gaussian ensemble, are optimal in minimizing the number of observations needed for recovery, albeit at a high computational cost. Sparse matrices, although computationally advantageous, may require more observations due to decreased statistical efficiency.

From a practical standpoint, understanding these trade-offs is vital for applications in compressive sensing, signal denoising, and network communication where resource constraints are present. Theoretical implications also abound, as the paper raises questions about potential improvements in sparse measurement designs that could approach the efficiency of dense ensembles.

Looking toward future developments, this research opens pathways for designing optimal measurement matrices that balance computational feasibility with statistical accuracy. Moreover, exploring the effectiveness of various recovery algorithms under these theoretical limits can further extend our understanding of sparse signal processing.

Overall, the paper significantly contributes to the domain of sparse signal recovery by refining theoretical limits and illuminating how sparsity in measurement matrices can affect recovery efficacy.