A Geometric Analysis of Phase Retrieval (1602.06664v3)

Abstract: Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of $m$ measurements, $y_k = |\mathbf a_k^* \mathbf x|$ for $k = 1, \dots, m$, is it possible to recover $\mathbf x \in \mathbb{C}^n$ (i.e., length-$n$ complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines, and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretical explanations. In this paper, we take a step towards bridging this gap. We prove that when the measurement vectors $\mathbf a_k$'s are generic (i.i.d. complex Gaussian) and the number of measurements is large enough ($m \ge C n \log^3 n$), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) there are no spurious local minimizers, and all global minimizers are equal to the target signal $\mathbf x$, up to a global phase; and (2) the objective function has a negative curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.

• The paper establishes that the nonconvex formulation for phase retrieval has no spurious local minima, ensuring global optimality up to an inherent phase ambiguity.
• It demonstrates that negative curvature at saddle points facilitates efficient escape from non-optimal solutions during iterative optimization.
• Empirical and theoretical analyses reveal that local strong convexity near the target enables robust convergence with a minimal number of measurements.

A Geometric Analysis of Phase Retrieval

The paper "A Geometric Analysis of Phase Retrieval" by Ju Sun, Qing Qu, and John Wright investigates the generalized phase retrieval (GPR) problem, a crucial task in fields such as crystallography and optical imaging. The authors address the challenge of recovering an $n$-dimensional complex vector $x$ from the magnitudes of its projections on known vectors, up to a global phase ambiguity.

Key Insights

The work provides a new perspective on the geometric properties of a natural nonconvex least-squares formulation for GPR. This formulation is given by:

$\mini_{z\in \mathbb{C}^n} \frac{1}{2m} \sum_{k=1}^m (y_k^2 - | a_k^* z|^2 )^2.$

Under the assumption that the measurement vectors are i.i.d. complex Gaussian and numerous enough (specifically, $m \ge C n \log^3 n$), the paper claims a structured landscape for the objective function. This landscape is characterized by:

1. Absence of Spurious Local Minima: All local minimizers are global, resolving ambiguities up to a global phase.
2. Negative Curvature at Saddle Points: The Hessian exhibits negative curvature in certain directions around saddle points, facilitating escape from local non-optimal solutions.
3. Strong Convexity Near the Target: Local strong convexity in the vicinity of the true solution ensures convergence to a global solution.

Theoretical and Practical Implications

Theoretically, these results bridge the gap between the empirical success of nonconvex methods and the lack of comprehensive theoretical guarantees. Practically, this structure implies that a variety of iterative optimization methods can achieve global optimum without requiring sophisticated initialization strategies. The paper corroborates these theoretical claims by analyzing a second-order trust-region method, proving its efficiency in converging to a global minimizer.

Numerical Experiments

Empirical results suggest that $m = 7n$ measurements suffice for reliable recovery, although theoretically, results are guaranteed when $m \ge C n \log^3 n$. This is an interesting observation as it hints towards potential improvements in theoretical sample complexity bounds.

Future Directions

While this work focuses on Gaussian measurements, extending this geometric analysis to structured measurement schemes like those found in practical settings (e.g., coded diffraction models) remains an open challenge. Additionally, applying similar geometric insights to other nonconvex problems in machine learning and signal processing could yield significant dividends.

Conclusion

This paper makes substantial progress in understanding the phase retrieval problem through the lens of geometric analysis of nonconvex landscapes. These insights not only clarify theoretical aspects but also pave the way for developing efficient, initialization-free algorithms for phase retrieval and related inverse problems in applied mathematics and engineering.