Special Seminar in Machine Learning and Optimization Optimization methods for inverse problems: The times they are a-changin
Sparsity-based methods for inverse problems gained widespread popularity in the 2000s when compressed-sensing theory provided groundbreaking insights on sparse recovery from randomized measurements. However, this theory does not explain the empirical success of sparse-recovery techniques in problems where the measurements are deterministic and structured. In the first half of this talk we will present a theory of sparse recovery for deterministic measurement operators relevant to optics, electroencephalography, quantitative magnetic-resonance imaging, and signal processing. In the second half we will illustrate the potential of deep neural networks in this domain with an application to super-resolution of line spectra. Deep learning techniques achieve remarkable empirical performance for a variety of inverse problems, but the mechanisms they implement are shrouded in mystery. We will conclude by showing that a local linear-algebraic analysis of a network trained for image denoising makes it possible to visualize some of the learned mechanisms, and reveals intriguing connections to traditional methodology.