On Learning and Generalization to Solve Inverse Problem of Electrophysiological Imaging

In this dissertation, we are interested in solving a linear inverse problem: inverse electrophysiological (EP) imaging, where our objective is to computationally reconstruct personalized cardiac electrical signals based on body surface electrocardiogram (ECG) signals. EP imaging has shown promise in the diagnosis and treatment planning of cardiac dysfunctions such as atrial flutter, atrial fibrillation, ischemia, infarction and ventricular arrhythmia. Towards this goal, we frame it as a problem of learning a function from the domain of measurements to signals. Depending upon the assumptions, we present two classes of solutions: 1) Bayesian inference in a probabilistic graphical model, 2) Learning from samples using deep networks. In both of these approaches, we emphasize on learning the inverse function with good generalization ability, which becomes a main theme of the dissertation. In a Bayesian framework, we argue that this translates to appropriately integrating different sources of knowledge into a common probabilistic graphical model framework and using it for patient specific signal estimation through Bayesian inference. In learning from samples setting, this translates to designing a deep network with good generalization ability, where good generalization refers to the ability to reconstruct inverse EP signals in a distribution of interest (which could very well be outside the sample distribution used during training). By drawing ideas from different areas like functional analysis (e.g. Fenchel duality), variational inference (e.g. Variational Bayes) and deep generative modeling (e.g. variational autoencoder), we show how we can incorporate different prior knowledge in a principled manner in a probabilistic graphical model framework to obtain a good inverse solution with generalization ability. Similarly, to improve generalization of deep networks learning from samples, we use ideas from information theory (e.g. information bottleneck), learning theory (e.g. analytical learning theory), adversarial training, complexity theory and functional analysis (e.g. RKHS). We test our algorithms on synthetic data and real data of the patients who had undergone through catheter ablation in clinics and show that our approach yields significant improvement over existing methods. Towards the end of the dissertation, we investigate general questions on generalization and stabilization of adversarial training of deep networks and try to understand the role of smoothness and function space complexity in answering those questions. We conclude by identifying limitations of the proposed methods, areas of further improvement and open questions that are specific to inverse electrophysiological imaging as well as broader, encompassing theory of learning and generalization

Explanation Uncertainty with Decision Boundary Awareness

Post-hoc explanation methods have become increasingly depended upon for understanding black-box classifiers in high-stakes applications, precipitating a need for reliable explanations. While numerous explanation methods have been proposed, recent works have shown that many existing methods can be inconsistent or unstable. In addition, high-performing classifiers are often highly nonlinear and can exhibit complex behavior around the decision boundary, leading to brittle or misleading local explanations. Therefore, there is an impending need to quantify the uncertainty of such explanation methods in order to understand when explanations are trustworthy. We introduce a novel uncertainty quantification method parameterized by a Gaussian Process model, which combines the uncertainty approximation of existing methods with a novel geodesic-based similarity which captures the complexity of the target black-box decision boundary. The proposed framework is highly flexible; it can be used with any black-box classifier and feature attribution method to amortize uncertainty estimates for explanations. We show theoretically that our proposed geodesic-based kernel similarity increases with the complexity of the decision boundary. Empirical results on multiple tabular and image datasets show that our decision boundary-aware uncertainty estimate improves understanding of explanations as compared to existing methods