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 $\{\mathrm{sign}(|\mathbf{a}_i^\top\mathbf{x}|-\tau)\}_{i=1}^m$ 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. To show this, we utilize a delicate covering argument and derive tight concentration bounds for the directional gradients by properly conditioning on the index set of phaseless hyperplane separations, which may be of independent interests and useful for other related problems.

翻译：本文研究了1比特相位恢复的样本复杂度，并开发了高效最优算法：从$m$个由标准高斯向量$\mathbf{a}_i$生成的无相位比特$\{\mathrm{sign}(|\mathbf{a}_i^\top\mathbf{x}|-\tau)\}_{i=1}^m$中恢复信号$\mathbf{x}\in\mathbb{R}^n$。通过研究无相位版本的随机超平面剖分，我们证明（约束）汉明距离最小化能够以误差$\mathcal{O}((n/m)\log(m/n))$均匀恢复所有欧几里得范数远离零和无穷大的无结构信号，当限制为$k$-稀疏信号时误差为$\mathcal{O}((k/m)\log(mn/k^2))$。两种误差率在信息论意义下均达到最优（对数因子除外）。值得注意的是，稀疏恢复的最优速率与1比特压缩感知相匹配，这表明对于1比特压缩感知而言，相位信息并非必要。我们还开发了能够达到这些误差率的1比特（稀疏）相位恢复高效算法。具体而言，我们证明当通过谱方法初始化时，针对单侧$\ell_1$损失的（阈值化）梯度下降能线性收敛并达到近最优重构误差，其中无结构信号的样本复杂度为$\mathcal{O}(n)$，$k$-稀疏信号为$\mathcal{O}(k^2\log(n)\log^2(m/k))$。我们的证明基于高斯测量满足某种局部（限制性）近似可逆性条件的观察。为证明这一点，我们采用精细的覆盖论证，并通过恰当条件化无相位超平面分离的指标集，导出方向梯度的紧致集中界——这一方法可能具有独立价值，并适用于其他相关问题。