[Paper Review] Deterministic Compressed Sensing Matrices from Additive Character Sequences

Nam Yul Yu|arXiv (Cornell University)|Sep 30, 2010

Sparse and Compressive Sensing Techniques24 references3 citations

TL;DR

This paper presents a deterministic construction of compressed sensing matrices using additive character sequences over finite fields, achieving asymptotically optimal coherence for $N = K^2$ and enabling efficient implementation via linear feedback shift registers (LFSRs). The method ensures reliable sparse signal recovery with matching pursuit, outperforming random partial Fourier matrices in both noiseless and noisy conditions, particularly for sparsity levels up to $s \leq 7$.

ABSTRACT

Compressed sensing is a novel technique where one can recover sparse signals from the undersampled measurements. In this correspondence, a $K imes N$ measurement matrix for compressed sensing is deterministically constructed via additive character sequences. The Weil bound is then used to show that the matrix has asymptotically optimal coherence for $N=K^2$, and to present a sufficient condition on the sparsity level for unique sparse recovery. Also, the restricted isometry property (RIP) is statistically studied for the deterministic matrix. Using additive character sequences with small alphabets, the compressed sensing matrix can be efficiently implemented by linear feedback shift registers. Numerical results show that the deterministic compressed sensing matrix guarantees reliable matching pursuit recovery performance for both noiseless and noisy measurements.

Motivation & Objective

• To develop a deterministic compressed sensing matrix construction that avoids the high complexity and storage of random matrices.
• To leverage additive character sequences over finite fields to ensure theoretical guarantees on coherence and restricted isometry property (RIP).
• To enable efficient hardware implementation using linear feedback shift registers (LFSRs) with small alphabets.
• To demonstrate robust and reliable sparse signal recovery performance under both noiseless and noisy measurement conditions.
• To provide a sufficient condition on sparsity level for unique sparse recovery using the deterministic matrix.

Proposed method

• Construct a $K \times N$ measurement matrix by using additive character sequences as column vectors, where $N = K^2$ and $K = p^m$ for prime $p$.
• Employ the trace function $\mathrm{Tr}_1^m(x)$ over $\mathbb{F}_{p^m}$ to generate complex-valued sequences via $\chi(x) = \exp\left(j \frac{2\pi}{p} \mathrm{Tr}_1^m(x)\right)$.
• Apply the Weil bound to prove that the coherence of the matrix is asymptotically optimal, with $\mu \leq \frac{2}{\sqrt{p^m}}$ for $N = K^2$.
• Analyze the restricted isometry property (RIP) statistically by studying eigenvalue distributions of the Gram matrix $\mathbf{A}^*\mathbf{A}$, showing favorable behavior compared to Gaussian random matrices.
• Implement the matrix efficiently using linear feedback shift registers (LFSRs) by exploiting the linear structure of additive character sequences.
• Use matching pursuit with a maximum of 100 iterations for signal reconstruction, with success defined by $\|\mathbf{x} - \widehat{\mathbf{x}}\|_2^2 < 10^{-4}$ (noiseless) or $< 10^{-2}$ (noisy).

Experimental results

Research questions

• RQ1 Can additive character sequences over finite fields be used to construct deterministic compressed sensing matrices with provably optimal coherence?
• RQ2 What is the sufficient condition on the sparsity level $s$ for unique recovery of $s$-sparse signals using the proposed deterministic matrix?
• RQ3 How does the restricted isometry property (RIP) of the deterministic matrix compare to that of random matrices in terms of eigenvalue statistics?
• RQ4 Can the proposed matrix be efficiently implemented in practice using linear feedback shift registers (LFSRs) with small alphabets?
• RQ5 How does the matching pursuit recovery performance of the deterministic matrix compare to that of randomly chosen partial Fourier matrices under noise and noiseless conditions?

Key findings

• The deterministic compressed sensing matrix achieves asymptotically optimal coherence for $N = K^2$, with coherence bounded by $\mu \leq \frac{2}{\sqrt{p^m}}$.
• For $K = 81$ and $N = 6561$, the matrix achieves over 99% successful recovery rate for $s \leq 4$ and over 95% for $s \leq 7$, indicating the theoretical sufficient condition is conservative.
• The matrix outperforms randomly chosen partial Fourier matrices in matching pursuit recovery performance under both noiseless and noisy conditions.
• In the presence of noise, the matrix maintains stable and noise-resilient recovery performance at high SNR, with success defined by $\|\mathbf{x} - \widehat{\mathbf{x}}\|_2^2 < 10^{-2}$.
• The eigenvalue statistics of the Gram matrix $\mathbf{A}^*\mathbf{A}$ show that the deterministic matrix has better RIP behavior than Gaussian random matrices.
• The matrix can be efficiently implemented using LFSRs, as each column is generated by a linear recurrence, enabling low-complexity hardware deployment with small alphabet size (e.g., $\mathbb{F}_3$).