[論文レビュー] Algorithmic Guarantees for Inverse Imaging with Untrained Network Priors

Gauri Jagatap, Chinmay Hegde|arXiv (Cornell University)|Jun 20, 2019

Sparse and Compressive Sensing Techniques参考文献 37被引用数 51

ひとこと要約

tldr: The paper introduces a theory and algorithms for using untrained deep network priors (deep decoder) to solve linear and non-linear inverse imaging problems, with convergence guarantees for projected gradient descent and phase retrieval settings.

ABSTRACT

Deep neural networks as image priors have been recently introduced for problems such as denoising, super-resolution and inpainting with promising performance gains over hand-crafted image priors such as sparsity and low-rank. Unlike learned generative priors they do not require any training over large datasets. However, few theoretical guarantees exist in the scope of using untrained neural network priors for inverse imaging problems. We explore new applications and theory for untrained neural network priors. Specifically, we consider the problem of solving linear inverse problems, such as compressive sensing, as well as non-linear problems, such as compressive phase retrieval. We model images to lie in the range of an untrained deep generative network with a fixed seed. We further present a projected gradient descent scheme that can be used for both compressive sensing and phase retrieval and provide rigorous theoretical guarantees for its convergence. We also show both theoretically as well as empirically that with deep network priors, one can achieve better compression rates for the same image quality compared to hand crafted priors.

研究の動機と目的

• Motivate the use of untrained deep neural networks as image priors for inverse imaging tasks.
• Develop a theoretical framework (REC variant) for range spaces of untrained networks to ensure unique recovery.
• Propose and analyze projected gradient descent methods that converge when using untrained priors in CS and phase retrieval.
• Extend the untrained-prior approach to compressive phase retrieval and provide convergence guarantees.
• Compare against hand-crafted priors and demonstrate empirical improvements in reconstruction quality.

提案手法

• Model images as lying in the range of an untrained deep network G(w;z) with fixed seed z.
• Define a set (S,γ,β)-REC for the measurement matrix A to guarantee recovery.
• Develop Net-PGD: a projected gradient descent algorithm with a projection onto the range S via optimizing w to fit an intermediate x.
• Prove convergence guarantees for Net-PGD under (S,1−α,1+α)-REC with high probability for Gaussian A.
• Extend to compressive phase retrieval with a two-step Net-PGD that includes phase estimation and a projection step.
• Base network architecture on an under-parameterized decoder (Deep Decoder) to ensure meaningful bounds and avoid fitting noise.

実験結果

リサーチクエスチョン

• RQ1 Can untrained neural network priors provide unique and stable recovery for linear compressive sensing with Gaussian measurements?
• RQ2 Do projected gradient descent schemes converge when the image prior is an untrained deep network?
• RQ3 Can untrained priors be effectively used for compressive phase retrieval with convergence guarantees?
• RQ4 How does the sample complexity depend on network parameters under the REC framework?
• RQ5 Do untrained priors offer superior reconstruction quality compared to hand-crafted priors in CS and CPR?

主な発見

• A new variant of the Restricted Eigenvalue Condition (REC) for the range of untrained network priors is established, enabling recovery guarantees.
• Net-PGD converges linearly to the true image under (S,1−α,1+α)-REC for linear compressive sensing with Gaussian measurements.
• Net-PGD extends to compressive phase retrieval with convergence guarantees under suitable initialization and REC assumptions.
• Empirical results show Net-GD and Net-PGD outperform traditional methods and other priors in reconstruction quality for MNIST and CelebA datasets.
• The framework emphasizes under-parameterized deep priors (Deep Decoder) to ensure meaningful bounds and prevent fitting noise.