Stability and accuracy in compressive sensing and deep learning

Abstract

There are currently two paradigm shifts happening in society and scientific computing: (1) Artificial Intelligence (AI) is replacing humans in problem solving, and, (2) AI is replacing the standard algorithms in computational science and engineering. Since reliable numerical calculations are paramount, algorithms for computational science are traditionally based on two pillars: accuracy and stability. Notably, this is true for image reconstruction, which is a mainstay of computational science, providing fundamental tools in medical, scientific and industrial imaging. In this thesis, we demonstrate that the stability pillar is typically absent in current deep learning and AI-based algorithms for image reconstruction, and we present a solution to why this phenomenon occurs for AI-based methods applied both to image reconstruction and to classification in general. This raises two fundamental questions: how reliable are such algorithms when applied in society, and do AI-based algorithms have the unavoidable Achilles heel of instability? We investigate these phenomena, and we introduce a framework designed to demonstrate, investigate and ultimately answer these fundamental questions.