Optimally Sample-Efficient Phase Retrieval with Deep Generative Models

Hand, Paul; Leong, Oscar; Voroninski, Vladislav

doi:10.1109/sampta45681.2019.9030906

Cited by 10 publications

(21 citation statements)

References 8 publications

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“…Non-linear measurement models. While Theorem 1 concerns linear observation models, analogous guarantees have been provided for a variety of non-linear measurement models, including 1-bit observations [85], [105], [69], spiked matrix models [7], [28], phase retrieval [84], [53], principal component analysis [87], and general single-index models [88], [83], [86]. While these each come with their own challenges, the intuition behind their associated results is often similar to that discussed above for the linear model, with the m = O k log Lr δ scaling typically remaining.…”

Section: E Further Developmentsmentioning

confidence: 98%

“…In the case of phase retrieval with random weights and expansivity, solving the optimization problem min z ∥y − |AG(z)|∥ 2 2 allows for signal recovery with m being proportional to k (ignoring the n and d dependence) [53]. This dependence is information-theoretically optimal, and it is noteworthy that it is attained with an efficient algorithm under random generative priors.…”

Section: E Further Developmentsmentioning

confidence: 99%

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Theoretical Perspectives on Deep Learning Methods in Inverse Problems

Scarlett

Heckel

Rodrigues

et al. 2022

IEEE J. Sel. Areas Inf. Theory

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In recent years, there have been significant advances in the use of deep learning methods in inverse problems such as denoising, compressive sensing, inpainting, and super-resolution. While this line of works has predominantly been driven by practical algorithms and experiments, it has also given rise to a variety of intriguing theoretical problems. In this paper, we survey some of the prominent theoretical developments in this line of works, focusing in particular on generative priors, untrained neural network priors, and unfolding algorithms. In addition to summarizing existing results in these topics, we highlight several ongoing challenges and open problems.

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Section: E Further Developmentsmentioning

confidence: 98%

Section: E Further Developmentsmentioning

confidence: 99%

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

Scarlett

Heckel

Rodrigues

et al. 2022

IEEE J. Sel. Areas Inf. Theory

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show abstract

“…Convergent methods have only been proposed for restricted uses cases. In compressed sensing problems with Gaussian measurement matrices, one can show that the objective function has a few critical points and design an algorithm to find the global optimum [13,18]. With a prior given by a VAE, and under technical hypothesis on the encoder and the VAE architecture, the Joint Posterior Maximization with Autoencoding Prior (JPMAP) framework of [11] converges toward a minimizer of the joint posterior p(x, z|y) of the image x and latent z given the observation y. JPMAP is nevertheless only designed for VAEs of limited expressiveness, with a simple fixed gaussian prior distribution over the latent space.…”

Section: Deep Generative Models For Inverse Problemsmentioning

confidence: 99%

Inverse problem regularization with hierarchical variational autoencoders

Prost¹,

Houdard²,

Almansa³

et al. 2023

Preprint

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In this paper, we propose to regularize ill-posed inverse problems using a deep hierarchical variational autoencoder (HVAE) as an image prior. The proposed method synthesizes the advantages of i) denoiser-based Plug & Play approaches and ii) generative model based approaches to inverse problems. First, we exploit VAE properties to design an efficient algorithm that benefits from convergence guarantees of Plug-and-Play (PnP) methods. Second, our approach is not restricted to specialized datasets and the proposed PnP-HVAE model is able to solve image restoration problems on natural images of any size. Our experiments show that the proposed PnP-HVAE method is competitive with both SOTA denoiser-based PnP approaches, and other SOTA restoration methods based on generative models. All experiments can be reproduced using the code available at https://github.com/jprost76/ PnP-HVAE

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“…There exists a plethora of methods to incorporate sparsity in phase retrieval. This includes convex approaches (Ohlsson, Yang, Dong and Sastry 2012, Li and Voroninski 2013), thresholding strategies (Wang et al 2017, Yuan, Wang and Wang 2019), greedy algorithms (Shechtman, Beck and Eldar 2014), algebraic methods (Beinert and Plonka 2017) and tools from deep learning (Hand, Leong and Voroninski 2018, Kim and Chung 2019). In the following we briefly discuss a few selected techniques in more detail.…”

Section: Convex Optimizationmentioning

confidence: 99%

“…Hand et al (2018) propose an alternative approach to model signals with a small number of parameters, based on generative models. They suppose that the signal of interest is in the range of a deep generative neural network , where the generative model is a d -layer, fully connected, feed-forward neural network with random weights.…”

Section: Convex Optimizationmentioning

confidence: 99%

The numerics of phase retrieval

Fannjiang

Strohmer

2020

Acta Numerica

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Phase retrieval, i.e. the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications, such as X-ray crystallography, diffraction imaging, optics, quantum mechanics and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by new imaging modalities and novel mathematical concepts. As our scientific experiments produce larger and larger datasets and we aim for faster and faster throughput, it is becoming increasingly important to study the involved numerical algorithms in a systematic and principled manner. Indeed, the past decade has witnessed a surge in the systematic study of computational algorithms for phase retrieval. In this paper we will review these recent advances from a numerical viewpoint.

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Optimally Sample-Efficient Phase Retrieval with Deep Generative Models

Cited by 10 publications

References 8 publications

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

Inverse problem regularization with hierarchical variational autoencoders

The numerics of phase retrieval

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