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Regularization Of Inverse Problems

Uncertainty quantification (UQ) plays a pivotal role in reduction of uncertainties during both optimization and decision making processes. It can be applied to solve a variety of real-world applications in science and engineering. Bayesian approximation and ensemble learning techniques are two most widely-used UQ methods in the literature. In this regard, researchers have proposed different UQ methods and examined their performance in a variety of applications such as computer vision (e.g., self-driving cars and object detection), image processing (e.g., image restoration), medical image analysis (e.g., medical image classification and segmentation), natural language processing (e.g., text classification, social media texts and recidivism risk-scoring), bioinformatics, etc. This study reviews recent advances in UQ methods used in deep learning. Moreover, we also investigate the application of these methods in reinforcement learning (RL). Then, we outline a few important applications of UQ methods. Finally, we briefly highlight the fundamental research challenges faced by UQ methods and discuss the future research directions in this field.

A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges

5分で分かる!? 有名論文ナナメ読み：Jacot, Arthor, Gabriel, Franck and Hongler, Clement : Neural Tangent Kernel : Convergence and Generalization in Neural Networks

http://jultika.oulu.fi/files/nbnfi-fe2021090645179.pdf

How do we compare between hypotheses that are entirely consistent with observations? The marginal likelihood (aka Bayesian evidence), which represents the probability of generating our observations from a prior, provides a distinctive approach to this foundational question, automatically encoding Occam's razor. Although it has been observed that the marginal likelihood can overfit and is sensitive to prior assumptions, its limitations for hyperparameter learning and discrete model comparison have not been thoroughly investigated. We first revisit the appealing properties of the marginal likelihood for learning constraints and hypothesis testing. We then highlight the conceptual and practical issues in using the marginal likelihood as a proxy for generalization. Namely, we show how marginal likelihood can be negatively correlated with generalization, with implications for neural architecture search, and can lead to both underfitting and overfitting in hyperparameter learning. We also re-examine the connection between the marginal likelihood and PAC-Bayes bounds and use this connection to further elucidate the shortcomings of the marginal likelihood for model selection. We provide a partial remedy through a conditional marginal likelihood, which we show is more aligned with generalization, and practically valuable for large-scale hyperparameter learning, such as in deep kernel learning.

Bayesian Model Selection, the Marginal Likelihood, and Generalization

The deep image prior regularises under-specified image reconstruction problems by reparametrising the target image as the output of a CNN. We induce a prior over images by scoring CNN outputs using a classical image reconstruction regulariser. We translate this functional prior into weight space using a change of variables and propose an efficient linearised Laplace inference algorithm. Hyperparameters are optimised with Type-II MAP. We obtain pixelwise uncertainty estimates, which we show to be calibrated to the reconstruction error.

A Probabilistic Deep Image Prior over Image Space

Recent advances in reconstruction methods for inverse problems leverage powerful data-driven models, e.g., deep neural networks. These techniques have demonstrated state-of-the-art performances for several imaging tasks, but they often do not provide uncertainty on the obtained reconstruction. In this work, we develop a scalable, data-driven, knowledge-aided computational framework to quantify the model uncertainty via Bayesian neural networks. The approach builds on, and extends deep gradient descent, a recently developed greedy iterative training scheme, and recasts it within a probabilistic framework. Scalability is achieved by being hybrid in the architecture: only the last layer of each block is Bayesian, while the others remain deterministic, and by being greedy in training. The framework is showcased on one representative medical imaging modality, viz. computed tomography with either sparse view or limited view data, and exhibits competitive performance with respect to state-of-the-art benchmarks, e.g., total variation, deep gradient descent and learned primal-dual.

Quantifying Model Uncertainty in Inverse Problems via Bayesian Deep Gradient Descent

Image reconstruction methods based on deep neural networks have shown outstanding performance, equalling or exceeding the state-of-the-art results of conventional approaches, but often do not provide uncertainty information about the reconstruction. In this work we propose a scalable and efficient framework to simultaneously quantify aleatoric and epistemic uncertainties in learned iterative image reconstruction. We build on a Bayesian deep gradient descent method for quantifying epistemic uncertainty, and incorporate the heteroscedastic variance of the noise to account for the aleatoric uncertainty. We show that our method exhibits competitive performance against conventional benchmarks for computed tomography with both sparse view and limited angle data. The estimated uncertainty captures the variability in the reconstructions, caused by the restricted measurement model, and by missing information, due to the limited angle geometry.

Quantifying Sources of Uncertainty in Deep Learning-Based Image Reconstruction

Deep learning-based image reconstruction approaches have demonstrated impressive empirical performance in many imaging modalities. These approaches usually require a large amount of high-quality paired training data, which is often not available in medical imaging. To circumvent this issue we develop a novel unsupervised knowledge-transfer paradigm for learned reconstruction within a Bayesian framework. The proposed approach learns a reconstruction network in two phases. The first phase trains a reconstruction network with a set of ordered pairs comprising of ground truth images of ellipses and the corresponding simulated measurement data. The second phase fine-tunes the pretrained network to more realistic measurement data without supervision. By construction, the framework is capable of delivering predictive uncertainty information over the reconstructed image. We present extensive experimental results on low-dose and sparse-view computed tomography showing that the approach is competitive with several state-of-the-art supervised and unsupervised reconstruction techniques. Moreover, for test data distributed differently from the training data, the proposed framework can significantly improve reconstruction quality not only visually, but also quantitatively in terms of PSNR and SSIM, when compared with learned methods trained on the synthetic dataset only. 

Unsupervised knowledge-transfer for learned image reconstruction*

—Existing deep-learning based tomographic image reconstruction methods do not provide accurate estimates of reconstruction uncertainty, hindering their real-world deployment. This paper develops a method, termed as the linearised deep image prior (DIP), to estimate the uncertainty associated with reconstructions produced by the DIP with total variation regularisation (TV). Speciﬁcally, we endow the DIP with conjugate Gaussian-linear model type error-bars computed from a local linearisation of the neural network around its optimised parameters. To preserve conjugacy, we approximate the TV regulariser with a Gaussian surrogate. This approach provides pixel-wise uncertainty estimates and a marginal likelihood objective for hyperparameter optimisation. We demonstrate the method on synthetic data and real-measured high-resolution 2D µ CT data, and show that it provides superior calibration of uncertainty estimates relative to previous probabilistic formulations of the DIP. Our code is available at https://github.com/educating-dip/bayes dip.

Riccardo Barbano

Papers

A Probabilistic Deep Image Prior over Image Space

Quantifying Model Uncertainty in Inverse Problems via Bayesian Deep Gradient Descent

Quantifying Sources of Uncertainty in Deep Learning-Based Image Reconstruction

Unsupervised knowledge-transfer for learned image reconstruction*

Uncertainty Estimation for Computed Tomography with a Linearised Deep Image Prior