Optimal Quantized Compressed Sensing via Projected Gradient Descent

TL;DR

This work develops a unified framework for recovering structured signals from quantized measurements $\mathbf{y}=\mathcal{Q}(\mathbf{A}\mathbf{x}-\bm{\tau})$ by employing projected gradient descent on a one-sided $\ell_1$-loss and establishing a restricted approximate invertibility condition (RAIC) that guarantees convergence with error close to information-theoretic limits. The method applies broadly to 1-bit CS, dithered 1-bit CS, and dithered multi-bit CS, with extensions to low-rank matrix recovery; the analysis combines sharp concentration bounds, gradient clipping for multi-bit settings, and a novel product-embedding approach to achieve global convergence in the multi-bit regime. Information-theoretic bounds are integrated with algorithmic guarantees to show that PGD attains the optimal or near-optimal rates, including $\tilde{O}(\frac{k}{mL})$ for $k$-sparse signals and $\tilde{O}((\frac{k}{mL})^{1/3})$ for effectively sparse signals, as well as analogous results for low-rank matrices. The results offer an efficient, scalable alternative to intractable decoders like constrained Hamming distance minimization, providing practical near-optimal recovery in a broad class of quantized sensing models.

Abstract

This paper provides a unified treatment to the recovery of structured signals living in a star-shaped set from general quantized measurements $\mathcal{Q}(\mathbf{A}\mathbf{x}-\mathbfτ)$, where $\mathbf{A}$ is a sensing matrix, $\mathbfτ$ is a vector of (possibly random) quantization thresholds, and $\mathcal{Q}$ denotes an $L$-level quantizer. The ideal estimator with consistent quantized measurements is optimal in some important instances but typically infeasible to compute. To this end, we study the projected gradient descent (PGD) algorithm with respect to the one-sided $\ell_1$-loss and identify the conditions under which PGD achieves the same error rate, up to logarithmic factors. These conditions include estimates of the separation probability, small-ball probability and some moment bounds that are easy to validate. For multi-bit case, we also develop a complementary approach based on product embedding to show global convergence. When applied to popular models such as 1-bit compressed sensing with Gaussian $\mathbf{A}$ and zero $\mathbfτ$ and the dithered 1-bit/multi-bit models with sub-Gaussian $\mathbf{A}$ and uniform dither $\mathbfτ$, our unified treatment yields error rates that improve on or match the sharpest results in all instances. Particularly, PGD achieves the information-theoretic optimal rate $\tilde{O}(\frac{k}{mL})$ for recovering $k$-sparse signals, and the rate $\tilde{O}((\frac{k}{mL})^{1/3})$ for effectively sparse signals. For 1-bit compressed sensing of sparse signals, our result recovers the optimality of normalized binary iterative hard thresholding (NBIHT) that was proved very recently.

Figures (6)

• Figure 1: An example of $(L=4)$-level quantizer. Note that we can assume (\ref{['wolg']}) because, for instance in the right figure, we can always modify $\{q_i\}_{i=1}^4$ to $\{q_i^{new}\}_{i=1}^4.$ Similarly, in 1-bit case where $\Delta=2$, we can always assume $(q_1,q_2)=(-1,1)$ to fulfill (\ref{['wolg']}).
• Figure 2: The uniform quantizer $\mathcal{Q}_\delta$ and its saturated version $\mathcal{Q}_{\delta,4}$ under the resolution $\delta=1$.
• Figure 3: RAIC indicates closeness of the ideal step and the actual subgradient step.
• Figure 4: In 1bCS, PGD achieves error rates $\tilde{O}(\frac{k}{m})$ for $\mathbf{x}\in \Sigma^n_k$ (left), $\tilde{O}(\frac{\bar{r}(n_1+n_2)}{m})$ for $\mathbf{x}\in M^{n_1,n_2}_{\bar{r}}$ (middle), and $\tilde{O}((\frac{k}{m})^{1/3})$ for $\mathbf{x}\in \sqrt{k}\mathbb{B}_1^n\cap\mathbb{S}^{n-1}$ (right). In sparse recovery, we recover $k$-sparse $500$-dimensional signals under $m=c\cdot\text{ml}$ with $\text{ml}=400:200:1200$. In low-rank recovery, we recover rank-$\bar{r}$$25\times 25$ matrices under $m=c\cdot\text{ml}$ with $\text{ml}=400:200:1200$. In recovering effectively sparse signals, we test $300$-dimensional signals under $m=c\cdot\text{ml}$ with $\text{ml}=800:400:2400$.
• Figure 5: In D1bCS, PGD achieves error rates $\tilde{O}(\frac{k}{m})$ for $\mathbf{x}\in \Sigma^n_k$ (left), $\tilde{O}(\frac{\bar{r}(n_1+n_2)}{m})$ for $\mathbf{x}\in M^{n_1,n_2}_{\bar{r}}$ (middle), and $\tilde{O}((\frac{k}{m})^{1/3})$ for $\mathbf{x}\in \sqrt{k}\mathbb{B}_1^n\cap\mathbb{S}^{n-1}$ (right). In sparse recovery, we recover $k$-sparse $500$-dimensional signals under $m=c\cdot\text{ml}$ with $\text{ml}=400:200:1600$. In low-rank recovery, we recover rank-$\bar{r}$$25\times 25$ matrices under measurement number $m=c\cdot\text{ml}$ with $\text{ml}=600:200:1600$ with $\Lambda=1.5$. In recovering effectively sparse signals, we test $300$-dimensional signals under $m=c\cdot\text{ml}$ with $\text{ml}=800:400:2400$.

Theorems & Definitions (85)

• Theorem 1: Information-theoretic lower bound
• Remark 1
• Remark 2
• Remark 3
• Theorem 2: Quantized embedding property
• Theorem 3: Information-theoretic upper bound
• proof
• Remark 4
• Remark 5
• Definition 1: RAIC