Decomposition Methods for Large Scale LP Decoding

When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper we draw on decomposition methods from optimization theory, specifically the Alternating Directions Method of Multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a "two-slice" characterization of the geometry of the parity polytope, which is the convex hull of all codewords of a single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by ADMM and allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher signal-to-noise ratio than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the 49th Annual Allerton Conference, September 2011. This version to appear in IEEE Transactions on Information Theor

Evaluating local indirect addressing in SIMD proc essors

In the design of parallel computers, there exists a tradeoff between the number and power of individual processors. The single instruction stream, multiple data stream (SIMD) model of parallel computers lies at one extreme of the resulting spectrum. The available hardware resources are devoted to creating the largest possible number of processors, and consequently each individual processor must use the fewest possible resources. Disagreement exists as to whether SIMD processors should be able to generate addresses individually into their local data memory, or all processors should access the same address. The tradeoff is examined between the increased capability and the reduced number of processors that occurs in this single instruction stream, multiple, locally addressed, data (SIMLAD) model. Factors are assembled that affect this design choice, and the SIMLAD model is compared with the bare SIMD and the MIMD models

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

Learning to Predict Combinatorial Structures

The major challenge in designing a discriminative learning algorithm for predicting structured data is to address the computational issues arising from the exponential size of the output space. Existing algorithms make different assumptions to ensure efficient, polynomial time estimation of model parameters. For several combinatorial structures, including cycles, partially ordered sets, permutations and other graph classes, these assumptions do not hold. In this thesis, we address the problem of designing learning algorithms for predicting combinatorial structures by introducing two new assumptions: (i) The first assumption is that a particular counting problem can be solved efficiently. The consequence is a generalisation of the classical ridge regression for structured prediction. (ii) The second assumption is that a particular sampling problem can be solved efficiently. The consequence is a new technique for designing and analysing probabilistic structured prediction models. These results can be applied to solve several complex learning problems including but not limited to multi-label classification, multi-category hierarchical classification, and label ranking.Comment: PhD thesis, Department of Computer Science, University of Bonn (submitted, December 2009

Embedding multidimensional grids into optimal hypercubes

Let $G$ and $H$ be graphs, with $|V(H)|\geq |V(G)|$, and $f:V(G)\rightarrow V(H)$ a one to one map of their vertices. Let $dilation(f) = max\{ dist_{H}(f(x),f(y)): xy\in E(G) \}$, where $dist_{H}(v,w)$ is the distance between vertices $v$ and $w$ of $H$. Now let $B(G,H)$ = $min_{f}\{ dilation(f) \}$, over all such maps $f$. The parameter $B(G,H)$ is a generalization of the classic and well studied "bandwidth" of $G$, defined as $B(G,P(n))$, where $P(n)$ is the path on $n$ points and $n = |V(G)|$. Let $[a_{1}\times a_{2}\times \cdots \times a_{k} ]$ be the $k$-dimensional grid graph with integer values $1$ through $a_{i}$ in the $i$'th coordinate. In this paper, we study $B(G,H)$ in the case when $G = [a_{1}\times a_{2}\times \cdots \times a_{k} ]$ and $H$ is the hypercube $Q_{n}$ of dimension $n = \lceil log_{2}(|V(G)|) \rceil$, the hypercube of smallest dimension having at least as many points as $G$. Our main result is that $$B( [a_{1}\times a_{2}\times \cdots \times a_{k} ],Q_{n}) \le 3k,$$ provided $a_{i} \geq 2^{22}$ for each $1\le i\le k$. For such $G$, the bound $3k$ improves on the previous best upper bound $4k+O(1)$. Our methods include an application of Knuth's result on two-way rounding and of the existence of spanning regular cyclic caterpillars in the hypercube.Comment: 47 pages, 8 figure