Sobolev Duals for Random Frames and Sigma-Delta Quantization of Compressed Sensing Measurements

Abstract

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size $\delta$ is used to quantize $m$ measurements $y=\Phi x$ of a $k$-sparse signal $x\in {\mathbb{R}}^N$, where $\Phi$ satisfies the restricted isometry property, then the approximate recovery $x^#$ via ${\ell }_1$-minimization is within $O(\delta )$ of $x$. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an $r$th order $\Sigma \Delta$ quantization scheme with the same output alphabet is used to quantize $y$, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduction of the approximation error by a factor of $(m/k{)}^{(r-1/2)\alpha }$ for any $0<\alpha <1$, if $m{\gtrsim }_{r}k(\log N{)}^{1/(1-\alpha )}$. The result holds with high probability on the initial draw of the measurement matrix $\Phi$ from the Gaussian distribution, and uniformly for all $k$-sparse signals $x$ that satisfy a mild size condition on their supports.