09/22 Woomyoung Park (UMD), Multilevel polarization for nonbinary codes and parallel channels

ABSTRACT: We study polarization for nonbinary channels with input alphabet of size q = 2^r; r = 2,3,... Using Arikan’s successive cancellation decoding, we prove that the virtual channels that arise in the process of polarization converge to q-ary channels with capacity 1, 2,...,r bits, and that the total transmission rate approaches the symmetric capacity of the channel. The multilevel polarization arising in this transmission represents a new phenomenon which is well suited to the description of dependent parallel channels that are subject to fading according to the order of their indices.
This is a joint work with Alexander Barg.

10/06, 10/13  Prakash Narayan (UMD), Multiuser secrecy, data compression and transmission, I,II
ABSTRACT: Secrecy generation for multiuser ``source" and ``channel" models will be considered, and connections drawn to associated multiuser compression and transmission problems (without secrecy constraints). Open problems (especially those that flummoxed us) will be described. This talk is based on joint work with Imre Csiszar, and the contents are taken from the three attached papers.

References: ic-pn2004.pdf, ic-pn2008.pdf, ic-pn2011.pdf

10/20 Alexander Barg (UMD) Statistical RIP and sparse recovery

ABSTRACT:  We consider recovery of sparse signals from noisy linear observations. A well-known sufficient condition for stable and robust recovery is the Restricted Isometry Property of the sampling operator. The known constructions of RIP matrices stop short of achieving the optimal sketch dimension.
We propose a relaxation of the RIP condition that permits shorter sketches and prove error bounds for some popular estimators. We also briefly discuss a construction of matrices that satisfy the relaxed conditions.
Joint work with Arya Mazumdar.

Wei-Hsuan Yu (UMD) Introduction to frame theory
ABSTRACT: Frames are interesting because they provide decompositions in applications where bases could be a liability. Tight frames are valuable to ensure fast
convergence of such decomposition. Normalized frames guarantee control of the frame elements. Finite frames avoid the subtle and omnipresent approximation problems associated with the truncation of infinite frames. In this talk, I will present the theory of finite normalized tight frames (FNTFs). The main theorem is the characterization of all FNTFs in terms of the minima of a potential energy function, which was designed to measure the total orthogonality of a Bessel sequence. Examples of FNTFs abound, e.g., in R 3 the vertices of the Platonic solids and of a soccer ball are FNTFs.

Radu Balan (UMD)
ABSTRACT:  In this talk we present an algorithm for signal reconstruction from absolute value of frame coefficients. Then we compare its performance to the Cramer-Rao Lower Bound (CRLB) at high signal-to-noise ratio. To fix notations, assume {f_i; 1<= i <= m} is a spanning set (hence frame) in R^n. Given noisy measurements d_i=|<x,f_i>|^2+\nu_i, 1<= i<= m, the problem is to recover x\in R^n up to a global sign. In this talk the reconstruction algorithm solves a regularized
least squares criterion of the form I(x) = \sum_{i=1}^m ||<x,f_i>|^2-d_i|^2 + \lambda ||x||^2 This criterion is modified in the following way: 1) the vector x is
replaced by a nxr matrix L; 2) the criterion is augmented to allow an iterative procedure. Once the matrix L has been obtained, an estimate for x is obtained through an SDV factorization.