Efficient Representations of Signals in Nonlinear Signal Processing with Applications to Inverse Problems

The focus of this thesis is the construction and analysis of efficient representations in nonlinear signal processing, and the applications of these structures to inverse problems in a variety of fields. The work is composed of three major sections, each associated with a different form of data: (1) Regression and Distance Estimation on Graphs and Riemannian Manifolds. (2) Instantaneous Time-Frequency Analysis via Synchrosqueezing. (3) Multiscale Dictionaries of Slepian Functions on the Sphere.