ASPIRE: Iterative Amortized Posterior Inference for Bayesian Inverse Problems

TL;DR

ASPIRE introduces an iterative approach to improve amortized Bayesian posterior inference, combining the speed of amortized methods with the accuracy of non-amortized techniques, especially useful in complex inverse problems.

Contribution

The paper proposes ASPIRE, a novel iterative method that enhances amortized variational inference for inverse problems without increasing online computational costs.

Findings

ASPIRE achieves higher fidelity posterior estimates than baseline methods.

The method maintains computational efficiency suitable for high-dimensional problems.

Validated on medical imaging and stylized problems, demonstrating practical effectiveness.

Abstract

Due to their uncertainty quantification, Bayesian solutions to inverse problems are the framework of choice in applications that are risk averse. These benefits come at the cost of computations that are in general, intractable. New advances in machine learning and variational inference (VI) have lowered the computational barrier by learning from examples. Two VI paradigms have emerged that represent different tradeoffs: amortized and non-amortized. Amortized VI can produce fast results but due to generalizing to many observed datasets it produces suboptimal inference results. Non-amortized VI is slower at inference but finds better posterior approximations since it is specialized towards a single observed dataset. Current amortized VI techniques run into a sub-optimality wall that can not be improved without more expressive neural networks or extra training data. We present a solution that enables iterative improvement of amortized posteriors that uses the same networks architectures and training data. The benefits of our method requires extra computations but these remain frugal since they are based on physics-hybrid methods and summary statistics. Importantly, these computations remain mostly offline thus our method maintains cheap and reusable online evaluation while bridging the approximation gap these two paradigms. We denote our proposed method ASPIRE - Amortized posteriors with Summaries that are Physics-based and Iteratively REfined. We first validate our method on a stylized problem with a known posterior then demonstrate its practical use on a high-dimensional and nonlinear transcranial medical imaging problem with ultrasound. Compared with the baseline and previous methods from the literature our method stands out as an computationally efficient and high-fidelity method for posterior inference.