[Paper Review] Error Decay of (almost) Consistent Signal Estimations from Quantized Random Gaussian Projections.

Laurent Jacques|arXiv (Cornell University)|May 30, 2014

Sparse and Compressive Sensing Techniques23 references4 citations

TL;DR

This paper establishes new error bounds for consistent signal reconstruction from quantized random Gaussian projections, showing that the worst-case $β_2$-error decays as $O(\frac{N}{M}\log\frac{M}{\sqrt{N}})$ for signals in the unit ball, matching known lower bounds up to a log factor. It further extends these results to $K$-sparse signals and demonstrates that relaxed consistency conditions still yield similar error decay rates under bounded quantization error.

ABSTRACT

This paper provides new error bounds on reconstruction methods for signals observed from quantized random sensing. Those signal estimation techniques guarantee a perfect matching between the available quantized data and a reobservation of the estimated signal under the same sensing model. Focusing on dithered uniform scalar quantization of resolution $\delta>0$, we prove first that, given a random Gaussian frame of $\mathbb R^N$ with $M$ vectors, the worst case $\ell_2$-error of consistent signal reconstruction decays with high probability as $O(\frac{N}{M}\log\frac{M}{\sqrt N})$ uniformly for all signals of the unit ball $\mathbb B^N \subset \mathbb R^N$. Up to a log factor, this matches a known lower bound in $\Omega(N/M)$. Equivalently, with a minimal number of frame coefficients behaving like $M = O(\frac{N}{\epsilon_0}\log(\frac{\sqrt N}{\epsilon_0}))$, any vectors in $\mathbb B^N$ with $M$ identical quantized projections are at most $\epsilon_0$ apart with high probability. Second, in the context of Quantized Compressed Sensing with $M$ random Gaussian measurements and under the same scalar quantization scheme, consistent reconstructions of $K$-sparse signals of $\mathbb R^N$ have a worst case error that decreases with high probability as $O(\frac{K}{M}\log\frac{MN}{\sqrt K^3})$ uniformly for all such signals. Finally, we show that the strict consistency condition can be slightly relaxed, e.g., allowing for a bounded level of error in the quantization process, while still guaranteeing a proximity between the original and the estimated signal. In particular, if this quantization error is of order $O(1)$ with respect to $M$, similar worst case error decays are reached for reconstruction methods adjusted to such an approximate consistency.

Motivation & Objective

• To derive tight error bounds for consistent signal estimation from quantized random Gaussian measurements.
• To analyze how the reconstruction error scales with the number of measurements $M$ and signal dimension $N$.
• To extend results to $K$-sparse signals in the context of Quantized Compressed Sensing.
• To investigate whether relaxing strict consistency improves robustness without sacrificing error decay rates.
• To establish error decay under bounded quantization error, showing approximate consistency still yields favorable convergence.

Proposed method

• Uses dithered uniform scalar quantization with resolution $\delta > 0$ to model quantized measurements.
• Applies a consistent reconstruction framework that enforces perfect match between estimated signal projections and observed quantized data.
• Employs probabilistic analysis on random Gaussian frames in $\mathbb{R}^N$ with $M$ vectors to derive high-probability error bounds.
• Derives worst-case $\ell_2$-error bounds using concentration inequalities and geometric arguments on the unit ball and sparse signal sets.
• Considers both full-dimensional signals and $K$-sparse signals, adjusting error bounds accordingly.
• Relaxes strict consistency by allowing bounded quantization error, and shows that similar error decay rates are preserved under adjusted reconstruction methods.

Experimental results

Research questions

• RQ1 How fast does the worst-case $\ell_2$-error decay for consistent signal reconstruction from quantized random Gaussian projections of signals in the unit ball?
• RQ2 What is the minimal number of measurements $M$ required to ensure that any two signals with identical quantized projections are within $\epsilon_0$ in $\ell_2$-norm with high probability?
• RQ3 How do error bounds scale for $K$-sparse signals under consistent reconstruction from quantized Gaussian measurements?
• RQ4 Can the strict consistency condition be relaxed while preserving favorable error decay rates?
• RQ5 What is the impact of bounded quantization error on the worst-case reconstruction error in consistent estimation?

Key findings

• The worst-case $\ell_2$-error for consistent reconstruction of signals in the unit ball $\mathbb{B}^N$ decays as $O(\frac{N}{M}\log\frac{M}{\sqrt{N}})$ with high probability.
• This error decay rate matches the known lower bound of $\Omega(\frac{N}{M})$ up to a logarithmic factor.
• With $M = O(\frac{N}{\epsilon_0}\log(\frac{\sqrt{N}}{\epsilon_0}))$ measurements, any two signals with identical quantized projections are at most $\epsilon_0$ apart in $\ell_2$-norm with high probability.
• For $K$-sparse signals, the worst-case $\ell_2$-error decays as $O(\frac{K}{M}\log\frac{MN}{\sqrt{K}^3})$ with high probability.
• Relaxing strict consistency to allow a bounded quantization error of order $O(1)$ with respect to $M$ still yields similar error decay rates.
• The proposed reconstruction methods maintain favorable error decay even under approximate consistency, demonstrating robustness to quantization imperfections.