Learned Proximal Networks for Unsupervised Inverse Problem Solving with Interpretable Priors

Core Concepts

Learned proximal networks (LPNs) are a new class of deep neural networks that exactly implement the proximal operator of a general learned function, enabling the recovery of the underlying data distribution's log-prior in an unsupervised manner. LPNs can be used to solve general inverse problems with convergence guarantees.

Abstract

The content discusses a new framework for learning proximal operators of general (potentially non-convex) functions using deep neural networks, termed learned proximal networks (LPNs). The key insights are:

LPNs are parameterized as gradients of convex functions, guaranteeing that they implement exact proximal operators. This allows for convergence guarantees when used in iterative optimization schemes like Plug-and-Play ADMM.

A new training strategy called proximal matching is proposed, which provably promotes the recovery of the log-prior of the true data distribution from i.i.d. samples, without requiring access to ground-truth proximal operators.

The ability to recover the underlying regularizer (log-prior) associated with the learned proximal operator provides interpretability and insights into the learned priors, which are demonstrated on synthetic and real-world datasets.

Experiments on image deblurring, sparse-view CT reconstruction, and compressed sensing show that LPNs achieve state-of-the-art performance while providing the additional benefit of interpretability.

Stats

The content does not contain any explicit numerical results or statistics. The key insights are conceptual and theoretical in nature.

Quotes

None.

Key Insights Distilled From

What's in a Prior? Learned Proximal Networks for Inverse Problems

by Zhenghan Fan... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2310.14344.pdf

Deeper Inquiries

How can the learned priors from LPNs be further leveraged for tasks like uncertainty quantification, robust optimization, and generative modeling

The learned priors from LPNs can be further leveraged for tasks like uncertainty quantification, robust optimization, and generative modeling in several ways.

Uncertainty Quantification: The learned priors can provide valuable insights into the uncertainty associated with the data distribution. By analyzing the variations in the learned prior, one can estimate the uncertainty in the model predictions. This can be crucial in decision-making processes where understanding the uncertainty is essential.

Robust Optimization: The non-convex priors learned by LPNs can be used to enhance robust optimization techniques. By incorporating the learned priors into the optimization process, one can improve the robustness of the solutions obtained, making them less sensitive to variations in the input data.

Generative Modeling: The learned priors can serve as a basis for generative modeling tasks. By leveraging the information encoded in the priors, one can generate new data samples that adhere to the learned distribution. This can be particularly useful in data augmentation, synthetic data generation, or even in creating realistic simulations.

Overall, the learned priors from LPNs offer a rich source of information that can be harnessed for a variety of tasks, enhancing the robustness, interpretability, and performance of machine learning models.

Can the proximal matching training strategy be extended to learn equivariant priors that respect the symmetries of the data

The proximal matching training strategy can indeed be extended to learn equivariant priors that respect the symmetries of the data. Equivariant learning aims to capture and leverage the inherent symmetries present in the data, leading to more efficient and effective models. By incorporating equivariance constraints into the proximal matching loss function, the LPNs can be trained to learn priors that exhibit the desired symmetries.
To extend the proximal matching strategy for equivariant priors:

Symmetry Constraints: Introduce constraints in the loss function that enforce equivariance properties, ensuring that the learned priors respect the symmetries present in the data.

Data Augmentation: Augment the training data with transformations that preserve the symmetries of interest. This can help the LPNs learn equivariant priors by exposing them to a diverse set of symmetric variations.

Regularization Techniques: Incorporate regularization terms that penalize deviations from equivariant behavior, encouraging the LPNs to learn priors that align with the symmetries of the data.

By extending the proximal matching strategy to learn equivariant priors, LPNs can be tailored to better capture the underlying structure and symmetries of the data, leading to more robust and efficient models.

What are the implications of learning non-convex priors for inverse problems, and how can this be exploited for better recovery performance

Learning non-convex priors for inverse problems has significant implications for the recovery performance and the overall effectiveness of the models.

Improved Recovery Performance: Non-convex priors can capture more complex and realistic data distributions that may not be adequately represented by convex priors. By learning non-convex priors, the models can better adapt to the intricacies and nuances of the data, leading to improved recovery performance and higher accuracy in inverse problem solutions.

Enhanced Flexibility: Non-convex priors offer greater flexibility in modeling diverse data distributions, allowing for more accurate and tailored solutions to inverse problems. This flexibility enables the models to capture a wider range of data variations and anomalies, enhancing their adaptability and robustness.

Exploration of Complex Structures: Non-convex priors can capture the complex underlying structures and relationships within the data, providing a more nuanced understanding of the data distribution. This can lead to more insightful interpretations of the learned models and facilitate deeper insights into the data generation process.

By exploiting non-convex priors for inverse problems, models can achieve higher accuracy, improved generalization, and a better understanding of the underlying data, ultimately enhancing the overall performance and effectiveness of the learning algorithms.