One-Bit Phase Retrieval: Optimal Rates and Efficient Algorithms

TL;DR

This work establishes tight information-theoretic limits and near-optimal algorithms for 1-bit phase retrieval with Gaussian measurements, recovering arbitrary signals in an annulus and k-sparse signals with rates matching known 1-bit compressed sensing bounds up to log factors. It introduces the phaseless hyperplane tessellation framework to analyze recovery and proves that constrained Hamming distance minimization achieves the optimal rates, while practical gradient-based methods with spectral initialization (GD-1bPR, BIHT-1bSPR) attain near-optimal error with near-linear convergence under feasible sample complexities. A key technical tool, PLL-AIC, ensures convergence of the proposed methods, and Gaussian matrices are shown to respect PLL-AIC, enabling rigorous guarantees. Spectral initializations, large- and small-distance regime analyses, and comparisons with 1-bit compressed sensing underpin the theoretical contributions, which are validated through comprehensive numerical experiments, including 1-bit phase retrieval with coded diffraction patterns on real images. The results highlight that phase information is non-essential for 1-bit compressed sensing in this phaseless setting and provide actionable algorithms with provable performance bounds, informing design of quantized sensing systems.

Abstract

In this paper, we study the sample complexity and develop efficient optimal algorithms for 1-bit phase retrieval: recovering a signal $\mathbf{x}\in\mathbb{R}^n$ from $m$ phaseless bits $\{\mathrm{sign}(|\mathbf{a}_i^\top\mathbf{x}|-τ)\}_{i=1}^m$ generated by standard Gaussian $\mathbf{a}_i$s. By investigating a phaseless version of random hyperplane tessellation, we show that (constrained) hamming distance minimization uniformly recovers all unstructured signals with Euclidean norm bounded away from zero and infinity to the error $\mathcal{O}((n/m)\log(m/n))$, and $\mathcal{O}((k/m)\log(mn/k^2))$ when restricting to $k$-sparse signals. Both error rates are shown to be information-theoretically optimal, up to a logarithmic factor. Intriguingly, the optimal rate for sparse recovery matches that of 1-bit compressed sensing, suggesting that the phase information is non-essential for 1-bit compressed sensing. We also develop efficient algorithms for 1-bit (sparse) phase retrieval that can achieve these error rates. Specifically, we prove that (thresholded) gradient descent with respect to the one-sided $\ell_1$-loss, when initialized via spectral methods, converges linearly and attains the near optimal reconstruction error, with sample complexity $\mathcal{O}(n)$ for unstructured signals and $\mathcal{O}(k^2\log(n)\log^2(m/k))$ for $k$-sparse signals. Our proof is based upon the observation that a certain local (restricted) approximate invertibility condition is respected by Gaussian measurements. Our results establish the major findings of (memoryless) 1-bit compressed sensing in a phaseless setting.

Figures (9)

• Figure 1: Graphical illustrations for the classical hyperplane tessellation (Left) and our proposed phaseless hyperplane tessellation (Right). In the left figure, the standard sphere $\{x_1^2+x_2^2=1\}$ in $\mathbb{R}^2$ is separated into 4 cells by two hyperplanes that pass through $(0,0)^\top$, providing geometric understanding of the standard 1-bit compressed sensing problem (e.g., plan2013oneplan2014dimensionoymak2015near). In the right figure, the annulus $\mathbbm{A}_{1,\sqrt{2}}=\{1\le x_1^2+x_2^2\le 2\}$ in $\mathbb{R}^2$ is separated into $4$ cells by two phaseless hyperplanes with a common shift $\tau = \sqrt{3/2}$, which reflects the geometric aspect of 1-bit phase retrieval.
• Figure 2: Phases are non-essential in solving 1-bit linear system (Left) and in 1-bit compressed sensing (Right). The reported data points are averaged over 50 independent trials. In the left figure we recover $\mathbf{x}\in \mathbb{R}^{30}$ uniformly distributed over $\mathbb{S}^{29}$ from $m=200:300:2000$ bits produced by Gaussian ensemble (the same below if not specified). The right figure is for the recovery of $\mathbf{x}\in\Sigma^{500,*}_3$ from $m=1000:500:3000$ bits, whereas we feed BIHT-1bSPR and SI-1bSPR with a (slightly) looser sparsity $k=4$ to simulate an actual setting where $k$ is not precisely known. The signals have support uniformly drawn from $\binom{500}{3}$ possibilities and non-zero entries uniformly distributed over $\mathbb{S}^2$.
• Figure 3: The impact of $\tau$ on 1bPR (Left) and 1bSPR (Right). We stick with the experimental settings in Figure \ref{['fig:inessential']} but fix $m=1500$. We test $\tau=0.1:0.1:3$ and two additional points $\tau=0.01,0.05$.
• Figure 4: Full Signal Reconstruction over $\mathbbm{A}_{\alpha,\beta}$ in 1-bit phase retrieval (Left) and 1-bit sparse phase retrieval (Right). Each data point is averaged over in $50$ independent trials. In the left figure, we recover $\mathbf{x}\in \mathbb{R}^{30}$ uniformly distributed over $\mathbbm{A}_{\alpha,\beta}$ from $m=500:500:2500$ phaseless bits under $\alpha=1$ and $\beta\in \{1,3,6,10,15\}$. In the right figure, we consider the same setting as in Figure \ref{['fig:inessential']}(Right) except that the signal $\mathbf{x}$ is rescaled to $\lambda \mathbf{x}$ with $\lambda$ uniformly distributed over $[\alpha,\beta]$, under $\alpha=1$ and $\beta\in\{1,1.2,1.5,2.5,3.5\}$.
• Figure 5: Recovering the $320\times 1280\times 3$ Stanford Main Quad image from phaseless bits produced by CDP with $L=32,~64$ random patterns. We run $50$ power method iterations for SI-1bPR and $100$ iterations for GD-1bPR, which involves $32\times 2 \times (50+100) = 9600,~64\times 2 \times (50+100) =19200$ FFTs (see candes2015phasechen2017solving) for each color band and completes in a few minutes.

Theorems & Definitions (103)

• Theorem 2.1: Local Binary Phaseless Embedding
• Theorem 2.2: Recovery via Hamming Distance Minimization
• Remark 2.1
• Theorem 2.3: Information-Theoretic Lower Bound
• Remark 2.2
• Remark 2.3
• Theorem 2.4: GD-1bPR is Near-Optimal
• Remark 2.4
• Remark 2.5
• Theorem 2.5: BIHT-1bSPR is Near-Optimal