Robust Instance Optimal Phase-Only Compressed Sensing

2024-08-12

arXiv.org

AI Summary

This work addresses robust recovery of structured signals in phase-only compressive sensing (PO-CS). To handle the nonlinearity inherent in complex Gaussian phase measurements, we propose a linearization technique that embeds phase-only measurements into the standard compressive sensing framework, enabling uniform instance-optimal reconstruction for all sparse signals on the unit sphere. We establish, for the first time in a nonlinear sensing setting, a unified and robust instance-optimal theory: with near-optimal sample complexity, the reconstruction error satisfies ‖x♯ − x‖₂ ≤ Cσₛ(x)₁/√s. The method exhibits strong robustness against both front-end and back-end additive noise, as well as sparse corruptions—enabling stable recovery even when sparse perturbations coexist with dense noise—and further guarantees exact recovery in the presence of sparse corruption. The theoretical performance matches that of linear compressive sensing, providing a novel paradigm for nonlinear sparse inverse problems.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Phase-only compressed sensing (PO-CS) is concerned with the recovery of structured signals from the phases of complex measurements. Recent results show that structured signals in the standard sphere $mathbb{S}^{n-1}$ can be exactly recovered from complex Gaussian phases, by recasting PO-CS as linear compressed sensing and then applying existing solvers such as basis pursuit. Known guarantees are either non-uniform or do not tolerate model error. We show that this linearization approach is more powerful than the prior results indicate. First, it achieves uniform instance optimality: Under complex Gaussian matrix with a near-optimal number of rows, this approach uniformly recovers all signals in $mathbb{S}^{n-1}$ with errors proportional to the model errors of the signals. Specifically, for sparse recovery there exists an efficient estimator $mathbf{x}^sharp$ and some universal constant $C$ such that $|mathbf{x}^sharp-mathbf{x}|_2le frac{Csigma_s(mathbf{x})_1}{sqrt{s}}~(forallmathbf{x}inmathbb{S}^{n-1})$, where $sigma_s(mathbf{x})_1=min_{mathbf{u}inSigma^n_s}|mathbf{u}-mathbf{x}|_1$ is the model error under $ell_1$-norm. Second, the instance optimality is robust to small dense disturbances and sparse corruptions that arise before or after capturing the phases. As an extension, we also propose to recast sparsely corrupted PO-CS as a linear corrupted sensing problem and show that this achieves perfect reconstruction of the signals. Our results resemble the instance optimal guarantees in linear compressed sensing and, to our knowledge, are the first results of this kind for a non-linear sensing scenario.

Problem

• Achieving uniform instance optimality in phase-only compressed sensing
• Analyzing robustness to noise in phase-only signal recovery
• Studying adversarial corruption effects on phase-only measurements

Innovation

• Linearization recasts PO-CS as linear compressed sensing
• Universal constant ensures uniform instance optimality
• Robustness to noise and adversarial corruption analyzed