Tight frame, Sparsity and Bregman algorithms

Applied and Computational Mathematics Seminar
Monday, April 5, 2010 - 13:00
1 hour (actually 50 minutes)
Skiles 255
Dep. of Math. UCLA
 Tight frame is a generalization of orthonormal basis. It  inherits most good properties of orthonormal basis but gains more  robustness to represent signals of intrests due to the redundancy. One can  construct tight frame systems under which signals of interests have sparse  representations. Such tight frames include translation invariant wavelet,  framelet, curvelet, and etc. The sparsity of a signal under tight frame systems has three different formulations, namely, the analysis-based sparsity, the synthesis-based one, and the balanced one between them. In this talk, we discuss Bregman algorithms for finding signals that are sparse under tight frame systems with the above three different formulations. Applications of our algorithms include image inpainting, deblurring, blind deconvolution, and cartoon-texture decomposition. Finally, we apply the linearized Bregman, one of the Bregman algorithms, to solve the problem of matrix completion, where we want to find a low-rank matrix from its incomplete entries. We view the low-rank matrix as a sparse vector under an adaptive linear transformation which depends on its singular vectors. It leads to a singular value thresholding (SVT) algorithm.