Robust Instance Optimal Phase-Only Compressed Sensing

TL;DR

This work studies phase-only compressed sensing (PO-CS) by leveraging a linearization to a real-valued sensing matrix and solving a quadratically constrained basis pursuit. It delivers a uniform instance-optimal guarantee, proving that a family of new sensing matrices satisfies RIP uniformly over arbitrary signal sets, and derives robustness bounds against bounded dense noise and sparse adversarial phase corruption. A key advance is an extended linearization that achieves exact recovery under sparse corruption, demonstrating that the detrimental effect of such corruption can be eliminated with an appropriate reformulation. Collectively, the results strengthen the theoretical foundation of PO-CS, providing uniform, robust recovery guarantees comparable to linear compressed sensing and suggesting practical pathways for reliable phase-only sensing in adverse environments.

Abstract

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.

Figures (3)

• Figure 1: Reconstruction errors under post-sign bounded by $\tau_0$
• Figure 2: Reconstruction errors under pre-sign bounded by $\tau_0$
• Figure 3: The impact of sparse phase corruption on $\|\mathbf{x}^\sharp-\mathbf{x}\|_2$

Theorems & Definitions (51)

• Proposition 2.1: see Thm. 2.1 in cai2013sparse
• Proposition 2.2: Thms. 4.3 & 4.6 in traonmilin2018stable
• Lemma 2.1: e.g., Thm. 2.10 in dirksen2021non
• Lemma 3.1: Deterministic error bound
• proof
• Theorem 3.1: RIP of a set of $\mathbf{A}_{\mathbf{z}}$
• Remark 3.1: Recovering $\mathcal{K}=\Sigma^{n,*}_s$ in chen2023uniform
• Remark 3.2: Arbitrary $\mathcal{K}\subset\mathbb{S}^{n-1}$
• Corollary 1: RIP of $\mathbf{A}_{\mathbf{z}}$ over approximately sparse signals
• Theorem 3.2: Uniform instance optimality