Solution of Linear Inverse Problems Using Flexible Dictionaries

• Activity Number: 98
• Type: Invited
• Date/Time: Monday, August 4, 2014 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Marianna Pensky*+
• Companies: University of Central Florida
• Keywords: Linear inverse problem ; Lasso ; adaptive estimation ; oracle inequality

Abstract

We consider solution of a general statistical linear inverse problem. Construction of an adaptive optimal solution for problems of this sort usually is either based on a singular value decomposition or relies on the knowledge of exact inverse images for a specific set of functions (like wavelets). The shortcoming of both approaches lies in the fact that, in many situations, neither the eigenbasis of the linear operator nor a standard dictionary (wavelets, trigonometric polynomials) constitutes an appropriate collection of functions for sparse representation of f. In the context of regression problems, there have been enormous amount of effort to recover an unknown function using a flexible, overcomplete dictionary. One of the most popular methods, Lasso and its versions, is based on minimizing empirical likelihood and, unfortunately, requires stringent assumptions on the dictionary, the, so called, compatibility conditions. While these conditions may be satisfied for the functions in the original dictionary, they usually do not hold for their images due to contraction imposed by the linear operator. In the talk, we show how one can go around compatibility conditions