Inverse Problems, Deep Learning, and Symmetry Breaking

• DOI: 10.48550/arxiv.2003.09077
• Publication Date: 2020-01-01
• Authors: Tayal, Kshitij; Lai, Chieh-Hsin; Kumar, Vipin; Sun, Ju
• Keywords: Signal Processing (eess.SP); FOS: Computer and information sciences; Computer Science - Machine Learning; Machine Learning (stat.ML); 02 engineering and technology; Numerical Analysis (math.NA); Machine Learning (cs.LG); Statistics - Machine Learning; Optimization and Control (math.OC); 0202 electrical engineering, electronic engineering, information engineering; FOS: Mathematics; FOS: Electrical engineering, electronic engineering, information engineering; Mathematics - Numerical Analysis; Electrical Engineering and Systems Science - Signal Processing; Mathematics - Optimization and Control

In many physical systems, inputs related by intrinsic system symmetries are mapped to the same output. When inverting such systems, i.e., solving the associated inverse problems, there is no unique solution. This causes fundamental difficulties for deploying the emerging end-to-end deep learning approach. Using the generalized phase retrieval problem as an illustrative example, we show that careful symmetry breaking on the training data can help get rid of the difficulties and significantly improve the learning performance. We also extract and highlight the underlying mathematical principle of the proposed solution, which is directly applicable to other inverse problems.