456 research outputs found
A Simple Message-Passing Algorithm for Compressed Sensing
We consider the recovery of a nonnegative vector x from measurements y = Ax,
where A is an m-by-n matrix whos entries are in {0, 1}. We establish that when
A corresponds to the adjacency matrix of a bipartite graph with sufficient
expansion, a simple message-passing algorithm produces an estimate \hat{x} of x
satisfying ||x-\hat{x}||_1 \leq O(n/k) ||x-x(k)||_1, where x(k) is the best
k-sparse approximation of x. The algorithm performs O(n (log(n/k))^2 log(k))
computation in total, and the number of measurements required is m = O(k
log(n/k)). In the special case when x is k-sparse, the algorithm recovers x
exactly in time O(n log(n/k) log(k)). Ultimately, this work is a further step
in the direction of more formally developing the broader role of
message-passing algorithms in solving compressed sensing problems
On the Complexity of Exact Maximum-Likelihood Decoding for Asymptotically Good Low Density Parity Check Codes: A New Perspective
The problem of exact maximum-likelihood (ML) decoding of general linear codes is well-known to be NP-hard. In this paper, we show that exact ML decoding of a class of asymptotically good low density parity check codes — expander codes — over binary symmetric channels (BSCs) is possible with an average-case polynomial complexity. This offers a new way of looking at the complexity issue of exact ML decoding for communication systems where the randomness in channel plays a fundamental central role. More precisely, for any bit-flipping probability p in a nontrivial range, there exists a rate region of non-zero support and a family of asymptotically good codes which achieve error probability exponentially decaying in coding length n while admitting exact ML decoding in average-case polynomial time. As p approaches zero, this rate region approaches the Shannon channel capacity region. Similar results can be extended to AWGN channels, suggesting it may be feasible to eliminate the error floor phenomenon associated with belief-propagation decoding of LDPC codes in the high SNR regime. The derivations are based on a hierarchy of ML certificate decoding algorithms adaptive to the channel realization. In this process, we propose an efficient O(n^2) new ML certificate algorithm based on the max-flow algorithm. Moreover, exact ML decoding of the considered class of codes constructed from LDPC codes with regular left degree, of which the considered expander codes are a special case, remains NP-hard; thus giving an interesting contrast between the worst-case and average-case complexities
Message passing for the coloring problem: Gallager meets Alon and Kahale
Message passing algorithms are popular in many combinatorial optimization
problems. For example, experimental results show that {\em survey propagation}
(a certain message passing algorithm) is effective in finding proper
-colorings of random graphs in the near-threshold regime. In 1962 Gallager
introduced the concept of Low Density Parity Check (LDPC) codes, and suggested
a simple decoding algorithm based on message passing. In 1994 Alon and Kahale
exhibited a coloring algorithm and proved its usefulness for finding a
-coloring of graphs drawn from a certain planted-solution distribution over
-colorable graphs. In this work we show an interpretation of Alon and
Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus
showing a connection between the two problems - coloring and decoding. This
also provides a rigorous evidence for the usefulness of the message passing
paradigm for the graph coloring problem. Our techniques can be applied to
several other combinatorial optimization problems and networking-related
issues.Comment: 11 page
Local Optimality Certificates for LP Decoding of Tanner Codes
We present a new combinatorial characterization for local optimality of a
codeword in an irregular Tanner code. The main novelty in this characterization
is that it is based on a linear combination of subtrees in the computation
trees. These subtrees may have any degree in the local code nodes and may have
any height (even greater than the girth). We expect this new characterization
to lead to improvements in bounds for successful decoding.
We prove that local optimality in this new characterization implies
ML-optimality and LP-optimality, as one would expect. Finally, we show that is
possible to compute efficiently a certificate for the local optimality of a
codeword given an LLR vector
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