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The focus of this thesis is the construction and analysis of efficientrepresentations in nonlinear signal processing, and the applications of thesestructures to inverse problems in a variety of fields. The work is composed ofthree major sections, each associated with a different form of data:  - Regression and Distance Estimation on Graphs and Riemannian Manifolds. -Instantaneous Time-Frequency Analysis via Synchrosqueezing. - MultiscaleDictionaries of Slepian Functions on the Sphere.

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