De-mixing problems in spectroscopic imaging often require finding sparse non-negative linear combinations of library functions to match observed data. Due to misalignment and uncertainty in data measurement, the known library functions may not represent the data as well as their proper deformations. To improve data adaptivity, we expand the library to one with a group structure and impose a structured sparsity constraint so that the coefficients for each group should be sparse or even 1-sparse. Since the expanded library is a highly coherent (redundant) dictionary, it is difficult to obtain good solutions using convex methods such as non-negative least squares (NNLS) or L1 norm minimization. We study efficient non-convex penalties such as the ratio/difference of L1 and L2 norms, as sparsity penalties to be added to the objective in NNLS-type models. We show an exact recovery theory of the sparsest solution by minimizing the ratio/difference norms under a uniformity condition. For solving the related unconstrained non-convex models, we develop a scaled gradient projection algorithm that requires solving a sequence of strongly convex quadratic programs.

Advisor: Qing Nie

We start with showing how rearrangement inequalities may be used in probabilistic contexts such as e.g. for obtaining bounds on survival probabilities in trapping models. This naturally motivates the need for a new rearrangement inequality which can be interpreted as involving symmetric rearrangements around infinity. After outlining the proof of this inequality we proceed to give some further applications to the volume of Lévy sausages as well as to capacities for Lévy processes.
(Joint work with P. Sousi and R. Sun)

Let G be a reductive group satisfying the Harish-Chandra condition defined over a totally real field F, E/F a finite cyclic extension of fields. With further assumptions on G, by constructing an explicit morphism between eigenvarities, we prove that every p-adic family of p-adic automorphic representations of G over F can be lifted to a family of p-adic automorphic representations of G over E such that , at every classical point, the lifting is just the classical weak base change lifting. The key ingredients in the theory are: (1) a twisted p-adic trace formula for G/E ; (2) a p-adic fundamental lemma and an equation between p-adic trace formula and twisted p-adic trace formula; (3) a second construction of a twisted eigenvariety of G/E.