Quantized Compressive Sensing

We study the average distortion introduced by scalar, vector, and entropy coded quantization of compressive sensing (CS) measurements. The asymptotic behavior of the underlying quantization schemes is either quantified exactly or characterized via bounds. We adapt two benchmark CS reconstruction algorithms to accommodate quantization errors, and empirically demonstrate that these methods significantly reduce the reconstruction distortion when compared to standard CS techniques.

💡 Research Summary

The paper “Quantized Compressive Sensing” investigates how quantization of compressive‑sensing (CS) measurements affects reconstruction quality and proposes reconstruction algorithms that explicitly account for quantization errors. The authors focus on average distortion rather than worst‑case analysis, and they study three quantization schemes: scalar quantization (both optimal non‑uniform and uniform), vector quantization, and entropy‑coded scalar quantization.

First, the authors review CS basics, the Basis Pursuit (BP) ℓ₁‑minimization method, and the Subspace Pursuit (SP) greedy algorithm. They then formalize quantization as a mapping from ℝ^m to a finite codebook C, define the mean‑squared error (MSE) distortion D_q = E‖Y – q(Y)‖², and introduce the distortion‑rate function D⁎(R) = inf_{C: (1/m)log₂|C| ≤ R} D(C). For scalar quantization they distinguish between optimal (non‑uniform) quantizers, designed via Lloyd’s algorithm, and low‑complexity uniform quantizers.

The core theoretical contribution is the asymptotic analysis of the distortion‑rate functions under two probabilistic models. In Model I, the measurement matrix Φ = (1/√m)A has i.i.d. sub‑Gaussian entries, and the K‑sparse signal x has i.i.d. sub‑Gaussian non‑zero components. Under this model, Theorem 1 shows that as the rate R → ∞ and (K, m, N) grow proportionally, the normalized distortion satisfies

 lim_{R→∞} lim_{K,m,N→∞} 2R·K·D⁎_{SQ}(R) = (π√3)/2,

and for uniform scalar quantization

 lim_{R→∞} lim_{K,m,N→∞} 2R·K·R·D⁎_{u,SQ}(R) = (4/3)·ln 2.

Thus optimal non‑uniform quantization yields roughly 1/R of the distortion of a uniform quantizer at high rates.

In Model II, the non‑zero entries of x are standard Gaussian, while Φ remains sub‑Gaussian. The authors introduce two matrix‑dependent constants: μ₁ = (1/N)∑{i,j} φ{ij}² (average column energy) and μ₂ = max_{i,T,|T|=K} ∑{j∈T} φ{ij}² (worst‑case column energy). Theorem 2 bounds the asymptotic distortion between (π√3)/2·μ₁ and (π√3)/2·μ₂ for optimal scalar quantization, and provides a similar lower bound for uniform quantization involving μ₁. These results highlight how the RIP‑type properties of Φ influence quantization‑induced distortion.

For vector quantization and entropy‑coded scalar quantization, exact closed‑form distortion‑rate functions are not derived; instead, the paper presents upper and lower bounds and argues that entropy coding can close the gap between uniform and optimal non‑uniform quantizers.

Recognizing that standard BP and SP ignore quantization errors, the authors modify both algorithms. In the quantization‑aware BP, the equality constraint y = Φx is replaced by a tolerance region defined by the quantization cell (typically a hypercube of side Δ). The ℓ₁ minimization is then solved with this relaxed constraint, effectively performing a constrained basis pursuit denoising. In the quantization‑aware SP, each iteration’s residual computation incorporates the known quantization error bound, and the support selection step uses the quantized residuals. The final least‑squares refinement also accounts for the quantization uncertainty.

Extensive simulations are performed with Gaussian measurement matrices (N = 1000, K = 10, m = 200) and various bit rates R = 2–6 bits per measurement. Results show that the quantization‑aware BP and SP achieve 3–7 dB higher reconstruction SNR compared with their naïve counterparts that treat quantized measurements as exact. Moreover, entropy‑coded non‑uniform scalar quantization provides an additional 1.5–2 dB gain over uniform quantization at the same average bit rate. The experiments also confirm that when μ₁≈μ₂ (i.e., Φ closely satisfies the RIP with small constants), the observed distortion approaches the theoretical lower bounds.

In conclusion, the paper establishes a rigorous average‑distortion framework for quantized compressive sensing, connects distortion performance to measurement matrix statistics, and delivers practical reconstruction algorithms that substantially improve performance in realistic, quantized acquisition systems. This work bridges the gap between information‑theoretic quantization analysis and algorithmic CS reconstruction, offering valuable guidance for the design of hardware‑constrained sensing devices.