6 research outputs found
Worst Configurations (Instantons) for Compressed Sensing over Reals: a Channel Coding Approach
We consider the Linear Programming (LP) solution of the Compressed Sensing
(CS) problem over reals, also known as the Basis Pursuit (BasP) algorithm. The
BasP allows interpretation as a channel-coding problem, and it guarantees
error-free reconstruction with a properly chosen measurement matrix and
sufficiently sparse error vectors. In this manuscript, we examine how the BasP
performs on a given measurement matrix and develop an algorithm to discover the
sparsest vectors for which the BasP fails. The resulting algorithm is a
generalization of our previous results on finding the most probable
error-patterns degrading performance of a finite size Low-Density Parity-Check
(LDPC) code in the error-floor regime. The BasP fails when its output is
different from the actual error-pattern. We design a CS-Instanton Search
Algorithm (ISA) generating a sparse vector, called a CS-instanton, such that
the BasP fails on the CS-instanton, while the BasP recovery is successful for
any modification of the CS-instanton replacing a nonzero element by zero. We
also prove that, given a sufficiently dense random input for the error-vector,
the CS-ISA converges to an instanton in a small finite number of steps. The
performance of the CS-ISA is illustrated on a randomly generated matrix. For this example, the CS-ISA outputs the shortest instanton (error
vector) pattern of length 11.Comment: Accepted to be presented at the IEEE International Symposium on
Information Theory (ISIT 2010). 5 pages, 2 Figures. Minor edits from previous
version. Added a new reference
Instanton-based Techniques for Analysis and Reduction of Error Floors of LDPC Codes
We describe a family of instanton-based optimization methods developed
recently for the analysis of the error floors of low-density parity-check
(LDPC) codes. Instantons are the most probable configurations of the channel
noise which result in decoding failures. We show that the general idea and the
respective optimization technique are applicable broadly to a variety of
channels, discrete or continuous, and variety of sub-optimal decoders.
Specifically, we consider: iterative belief propagation (BP) decoders, Gallager
type decoders, and linear programming (LP) decoders performing over the
additive white Gaussian noise channel (AWGNC) and the binary symmetric channel
(BSC).
The instanton analysis suggests that the underlying topological structures of
the most probable instanton of the same code but different channels and
decoders are related to each other. Armed with this understanding of the
graphical structure of the instanton and its relation to the decoding failures,
we suggest a method to construct codes whose Tanner graphs are free of these
structures, and thus have less significant error floors.Comment: To appear in IEEE JSAC On Capacity Approaching Codes. 11 Pages and 6
Figure