Phase-only compressed sensing (PO-CS) concerns the recovery of sparse signals from the phases of complex measurements. Recent results show that sparse signals in the standard sphere $\mathbb{S}^{n-1}$ can be exactly recovered from complex Gaussian phases by a linearization procedure, which recasts PO-CS as linear compressed sensing and then applies (quadratically constrained) basis pursuit to obtain $\mathbf{x}^\sharp$. This paper focuses on the instance optimality and robustness of $\mathbf{x}^{\sharp}$. First, we strengthen the nonuniform instance optimality of Jacques and Feuillen (2021) to a uniform one over the entire signal space. We show the existence of some universal constant $C$ such that $\|\mathbf{x}^\sharp-\mathbf{x}\|_2\le Cs^{-1/2}σ_{\ell_1}(\mathbf{x},Σ^n_s)$ holds for all $\mathbf{x}$ in the unit Euclidean sphere, where $σ_{\ell_1}(\mathbf{x},Σ^n_s)$ is the $\ell_1$ distance of $\mathbf{x}$ to its closest $s$-sparse signal. This is achieved by showing the new sensing matrices corresponding to all approximately sparse signals simultaneously satisfy RIP. Second, we investigate the estimator's robustness to noise and corruption. We show that dense noise with entries bounded by some small $τ_0$, appearing either prior or posterior to retaining the phases, increments $\|\mathbf{x}^\sharp-\mathbf{x}\|_2$ by $O(τ_0)$. This is near-optimal (up to log factors) for any algorithm. On the other hand, adversarial corruption, which changes an arbitrary $ζ_0$-fraction of the measurements to any phase-only values, increments $\|\mathbf{x}^\sharp-\mathbf{x}\|_2$ by $O(\sqrt{ζ_0\log(1/ζ_0)})$. The developments are then combined to yield a robust instance optimal guarantee that resembles the standard one in linear compressed sensing.

翻译：相位压缩感知（PO-CS）关注从复数测量的相位中恢复稀疏信号的问题。近期结果表明，标准球面 $\mathbb{S}^{n-1}$ 中的稀疏信号可通过线性化过程从复高斯相位中精确恢复，该过程将 PO-CS 重构为线性压缩感知问题，然后应用（二次约束的）基追踪来获得 $\mathbf{x}^\sharp$。本文重点研究 $\mathbf{x}^{\sharp}$ 的实例最优性与鲁棒性。首先，我们将 Jacques 和 Feuillen（2021）的非均匀实例最优性强化为在整个信号空间上的一致最优性。我们证明存在某个通用常数 $C$，使得 $\|\mathbf{x}^\sharp-\mathbf{x}\|_2\le Cs^{-1/2}σ_{\ell_1}(\mathbf{x},Σ^n_s)$ 对单位欧几里得球面中的所有 $\mathbf{x}$ 均成立，其中 $σ_{\ell_1}(\mathbf{x},Σ^n_s)$ 是 $\mathbf{x}$ 到其最近 $s$-稀疏信号的 $\ell_1$ 距离。这是通过证明对应于所有近似稀疏信号的新传感矩阵同时满足 RIP 来实现的。其次，我们研究了估计器对噪声与扰动的鲁棒性。我们证明，无论出现在保留相位之前或之后、其元素受某个小值 $τ_0$ 限制的稠密噪声，会使 $\|\mathbf{x}^\sharp-\mathbf{x}\|_2$ 增加 $O(τ_0)$。这对任何算法而言都是近乎最优的（忽略对数因子）。另一方面，对抗性扰动将任意 $ζ_0$ 比例的测量值更改为任意仅相位值，会使 $\|\mathbf{x}^\sharp-\mathbf{x}\|_2$ 增加 $O(\sqrt{ζ_0\log(1/ζ_0)})$。最后，我们将这些进展结合，得到了一个类似于线性压缩感知中标准形式的鲁棒实例最优性保证。