Phase Transitions in Phase-Only Compressed Sensing

2025-01-21

AI Summary

This work investigates the fundamental limits of compressive sensing for structured signals—specifically sparse vectors and low-rank matrices—using phase-only measurements. It establishes an equivalence between phase retrieval and linear compressive sensing, enabling reconstruction via basis pursuit. Methodologically, it leverages the Gaussian min-max theorem and signal-dependent descent cone analysis to derive precise asymptotic phase transition thresholds. The key contributions are: (i) a rigorous demonstration that the phase-only sensing phase transition threshold is strictly lower than that of conventional linear compressive sensing—refuting the earlier conjecture of their coincidence; (ii) closed-form expressions for the phase transitions in both sparse and low-rank settings; and (iii) for 1-sparse signals, a reduction in required measurements to approximately 68% of those needed in standard linear compressive sensing, yielding substantial sampling efficiency gains.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web measurements

Abstract

The goal of phase-only compressed sensing is to recover a structured signal $mathbf{x}$ from the phases $mathbf{z} = { m sign}(mathbf{Phi}mathbf{x})$ under some complex-valued sensing matrix $mathbf{Phi}$. Exact reconstruction of the signal's direction is possible: we can reformulate it as a linear compressed sensing problem and use basis pursuit (i.e., constrained norm minimization). For $mathbf{Phi}$ with i.i.d. complex-valued Gaussian entries, this paper shows that the phase transition is approximately located at the statistical dimension of the descent cone of a signal-dependent norm. Leveraging this insight, we derive asymptotically precise formulas for the phase transition locations in phase-only sensing of both sparse signals and low-rank matrices. Our results prove that the minimum number of measurements required for exact recovery is smaller for phase-only measurements than for traditional linear compressed sensing. For instance, in recovering a 1-sparse signal with sufficiently large dimension, phase-only compressed sensing requires approximately 68% of the measurements needed for linear compressed sensing. This result disproves earlier conjecture suggesting that the two phase transitions coincide. Our proof hinges on the Gaussian min-max theorem and the key observation that, up to a signal-dependent orthogonal transformation, the sensing matrix in the reformulated problem behaves as a nearly Gaussian matrix.

Problem

• Compressive Sensing
• Phase Information
• Signal Recovery

Innovation

• Phase-Only Compressive Sensing
• Measurement Efficiency
• Sparse Signal Recovery