Phase Retrieval Under a Generative Prior (1807.04261v1)

Abstract: The phase retrieval problem asks to recover a natural signal $y_0 \in \mathbb{R}^n$ from $m$ quadratic observations, where $m$ is to be minimized. As is common in many imaging problems, natural signals are considered sparse with respect to a known basis, and the generic sparsity prior is enforced via $\ell_1$ regularization. While successful in the realm of linear inverse problems, such $\ell_1$ methods have encountered possibly fundamental limitations, as no computationally efficient algorithm for phase retrieval of a $k$-sparse signal has been proven to succeed with fewer than $O(k^2\log n)$ generic measurements, exceeding the theoretical optimum of $O(k \log n)$. In this paper, we propose a novel framework for phase retrieval by 1) modeling natural signals as being in the range of a deep generative neural network $G : \mathbb{R}^k \rightarrow \mathbb{R}^n$ and 2) enforcing this prior directly by optimizing an empirical risk objective over the domain of the generator. Our formulation has provably favorable global geometry for gradient methods, as soon as $m = O(kd^2\log n)$, where $d$ is the depth of the network. Specifically, when suitable deterministic conditions on the generator and measurement matrix are met, we construct a descent direction for any point outside of a small neighborhood around the unique global minimizer and its negative multiple, and show that such conditions hold with high probability under Gaussian ensembles of multilayer fully-connected generator networks and measurement matrices. This formulation for structured phase retrieval thus has two advantages over sparsity based methods: 1) deep generative priors can more tightly represent natural signals and 2) information theoretically optimal sample complexity. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms sparse phase retrieval methods.

• The paper introduces the Deep Phase Retrieval (DPR) framework that leverages generative models to reconstruct signals with reduced sample complexity compared to traditional sparsity-based methods.
• It employs a gradient descent strategy to navigate non-convex landscapes and achieve efficient recovery from fewer quadratic measurements.
• Rigorous theoretical analysis and experiments on datasets like MNIST and CelebA validate DPR’s superiority, opening new avenues for advanced imaging and signal processing.

Analysis of "Phase Retrieval Under a Generative Prior"

The paper "Phase Retrieval Under a Generative Prior" introduces a novel approach to the phase retrieval problem by leveraging the capacity of generative models. The authors address the phase retrieval challenge of reconstructing a natural signal $y_0 \in \mathbb{R}^n$ from $m$ quadratic measurements while minimizing $m$. Traditionally, sparse priors have been employed using $\ell_1$ regularization, which though effective in linear inverse problems, has limitations in double the sample complexity compared to the theoretical optimum in phase retrieval scenarios. This research proposes a new methodology intended to leverage deep generative neural networks (GNNs) as a prior for phase retrieval.

Key Contributions

1. Deep Phase Retrieval (DPR) Framework: The authors propose a framework where natural signals are assumed to be in the range of a deep GNN, $G: \mathbb{R}^k \rightarrow \mathbb{R}^n$. This model enables encoding complex, natural data distributions more suitably than traditional sparsity-based methods.
2. Sample Complexity and Geometric Favorability:
   • The paper demonstrates that under specific conditions, the DPR framework models a benign global geometry that is favorable for efficient optimization strategies.
   • It is shown that DPR achieves sample complexity $m = O(kd^2\log n)$, which is more optimal than the limitations encountered when using sparsity priors.
3. Gradient Scheme and Empirical Validation: DPR employs a gradient descent approach which can escape unfavorable optimization basins. The empirical analysis supported by experiments on synthetic and real-world signals (e.g., MNIST and CelebA datasets) reveals that the proposed method surpasses existing sparse phase retrieval approaches, particularly at lower measurement regimes.
4. Structured Theoretical Analysis: The paper includes rigorous establishment of conditions under which the favorable landscape is achieved, highlighting the concentration properties of the objective under Gaussian ensembles for multilayer networks and measurement matrices.

Technical Implications

The paper opens up a significant theoretical implication for non-convex optimization in signal processing where conventional convex approaches show inefficacies. By using generative models, the paper presents the result that more expressive priors can diminish the sample gap that the sparsity assumption failed to address effectively.

Future Directions

The authors speculate potential applicability of the framework to broader scientific imaging fields and suggest a pathway for future work in exploring different classes of priors beyond those based on sparsity. Additionally, the research implies that with better training data and networks, the phase retrieval problem could be approached with even greater efficiency and accuracy. Expanding the understanding of this newly proposed geometric landscape could foster innovations in optimization algorithms design, further minimizing reliance on assumptions of data distributions.

In summary, "Phase Retrieval Under a Generative Prior" extends the capabilities of phase retrieval from relying on sparsity-based methods to utilizing deep learning-based models, thus reducing the redundancy in measurements and introducing more robust ways of solving complex imaging problems.