[Paper Review] Compressive sensing by white random convolution

Yin Xiang, LianLin Li+1 more|ArXiv.org|Sep 15, 2009

Sparse and Compressive Sensing Techniques22 references3 citations

TL;DR

This paper proposes a compressive sensing framework using white random convolution followed by fixed subsampling, demonstrating that sparse signals can be reliably recovered with high probability when the coherence between the signal basis and Fourier basis is bounded. The key result shows that m ≈ O(μ²S log(n/δ)^{3/2}) samples suffice for recovery with probability exceeding 1−δ, where μ is the coherence, S is sparsity, and n is signal dimension.

ABSTRACT

A different compressive sensing framework, convolution with white noise waveform followed by subsampling at fixed (not randomly selected) locations, is studied in this paper. We show that its recoverability for sparse signals depends on the coherence (denoted by mu) between the signal representation and the Fourier basis. In particular, an n-dimensional signal which is S-sparse in such a basis can be recovered with a probability exceeding 1-delta from any fixed m~O(mu^2*S*log(n/delta)^(3/2)) output samples of the random convolution.

Motivation & Objective

• Address the challenge of designing efficient, implementable compressive sensing systems with deterministic sampling patterns.
• Investigate whether white random convolution combined with fixed subsampling can achieve stable recovery of sparse signals.
• Characterize the conditions under which such a system ensures high-probability recovery of S-sparse signals in a given basis.
• Quantify the number of required samples in terms of coherence μ, sparsity S, signal dimension n, and failure probability δ.
• Provide theoretical guarantees for recovery using a framework that avoids random selection of sampling locations, enabling practical hardware implementation.

Proposed method

• Apply white noise waveform as a convolution kernel to the sparse signal, transforming it into a spread spectrum signal.
• Subsample the convolved signal at fixed, predetermined locations rather than using random sampling patterns.
• Model the measurement process as a linear transformation governed by the convolution kernel and fixed sampling matrix.
• Use coherence μ between the signal representation basis and the Fourier basis as a key parameter to bound recovery performance.
• Apply probabilistic analysis to derive sample complexity bounds under the assumption of bounded coherence.
• Leverage concentration inequalities and properties of random convolution to establish recovery guarantees with high probability.

Experimental results

Research questions

• RQ1 Can white random convolution followed by fixed subsampling achieve stable recovery of sparse signals in compressive sensing?
• RQ2 How does the coherence μ between the signal basis and the Fourier basis affect the required number of samples for recovery?
• RQ3 What is the minimal number of fixed subsampling points needed to ensure high-probability recovery of S-sparse signals?
• RQ4 Does the absence of random sampling locations compromise recovery performance, and if so, under what conditions can it still be guaranteed?
• RQ5 Can theoretical recovery guarantees be established for a deterministic sampling pattern using random convolution?

Key findings

• The proposed framework enables high-probability recovery of S-sparse signals from m ≈ O(μ²S log(n/δ)^{3/2}) fixed subsampling points.
• Recovery is guaranteed with probability exceeding 1−δ, where δ controls the failure rate.
• The coherence μ between the signal representation basis and the Fourier basis is the central parameter determining sample complexity.
• The method avoids random sampling patterns, making it suitable for practical hardware implementation.
• The theoretical bound scales favorably with sparsity S and logarithmically with signal dimension n and inverse failure probability δ.
• The framework maintains stable recovery performance even with deterministic sampling, provided the coherence μ is bounded.