🤖 AI Summary
This paper studies the 1-bit phase retrieval problem: reconstructing a real signal $mathbf{x} in mathbb{R}^n$—either dense or $k$-sparse—from $m$ 1-bit measurements $mathrm{sign}(|mathbf{a}_i^ op mathbf{x}| - au)$, where $mathbf{a}_i sim mathcal{N}(0,I_n)$. We establish the first information-theoretically optimal estimation error bounds, demonstrating that phase information is unnecessary in 1-bit compressed sensing. We propose a one-sided $ell_1$ gradient descent algorithm with spectral initialization, achieving linear convergence and near-optimal statistical accuracy. Key technical innovations include random hyperplane tessellation analysis, local restricted approximate invertibility characterization, and thresholded gradient updates. Our theory yields optimal error rates $O((n/m)log(m/n))$ for dense signals and $O((k/m)log(mn/k^2))$ for $k$-sparse signals. The algorithm attains sample complexities of $O(n)$ and $O(k^2 log n cdot log^2 (m/k))$, respectively—achieving both computational efficiency and statistical optimality.
📝 Abstract
In this paper, we study the sample complexity and develop efficient optimal algorithms for 1-bit phase retrieval: recovering a signal $mathbf{x}inmathbb{R}^n$ from $m$ phaseless bits ${mathrm{sign}(|mathbf{a}_i^ opmathbf{x}|- au)}_{i=1}^m$ 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 $mathcal{O}(k^2log(n)log^2(m/k))$ for $k$-sparse signals. Our proof is based upon the observation that a certain local (restricted) approximate invertibility condition is respected by Gaussian measurements.