Data-Driven Techniques for Inverse Problems: Priors, Measurements, and Decoding
Abstract:
The accurate resolution of an inverse problem hinges on the cohesive integration of three elements: prior knowledge, measurement mechanisms, and decoding algorithms. Each component plays a pivotal role in shaping the problem formulation, orchestrating data acquisition process, and extracting meaningful insights from observation.
The rise of deep neural networks has prompted a focused endeavor to transform the landscape of solving inverse problems through the utilization of data-driven strategies and deep learning methodologies. For example, for priors, deep generative models and untrained neural networks have been explored as avenues to capture the underlying structure of data distributions. For measurement mechanisms, training data has been employed in diverse ways to design measurement matrices, such as fine-tuning parametrized designs, selecting optimal subsampling patterns, and directly learning matrices. As for decoding, deep neural networks have been utilized to exploit large-scale datasets for training, facilitating the creation of end-to-end estimators capable of understanding complex relationships between inputs and outputs.
However, there are still several significant challenges unresolved, and in this dissertation, we aim to tackle several such challenges within each of the three aspects of inverse problems: prior knowledge, measurement mechanisms, and decoding algorithms.