Thermodynamics of the Binary Symmetric Channel

We study a hidden Markov process which is the result of a transmission of the binary symmetric Markov source over the memoryless binary symmetric channel. This process has been studied extensively in Information Theory and is often used as a benchmark case for the so-called denoising algorithms. Exploiting the link between this process and the 1D Random Field Ising Model (RFIM), we are able to identify the Gibbs potential of the resulting Hidden Markov process. Moreover, we obtain a stronger bound on the memory decay rate. We conclude with a discussion on implications of our results for the development of denoising algorithms

Hierarchical interpolative factorization for elliptic operators: integral equations

This paper introduces the hierarchical interpolative factorization for integral equations (HIF-IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIF-IE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higher-dimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIF-IE is compatible with geometric adaptivity and can handle both boundary and volume problems. MATLAB codes are freely available.Comment: 39 pages, 14 figures, 13 tables; to appear, Comm. Pure Appl. Mat

Exploring single-sample SNP and INDEL calling with whole-genome de novo assembly

Motivation: Eugene Myers in his string graph paper (Myers, 2005) suggested that in a string graph or equivalently a unitig graph, any path spells a valid assembly. As a string/unitig graph also encodes every valid assembly of reads, such a graph, provided that it can be constructed correctly, is in fact a lossless representation of reads. In principle, every analysis based on whole-genome shotgun sequencing (WGS) data, such as SNP and insertion/deletion (INDEL) calling, can also be achieved with unitigs. Results: To explore the feasibility of using de novo assembly in the context of resequencing, we developed a de novo assembler, fermi, that assembles Illumina short reads into unitigs while preserving most of information of the input reads. SNPs and INDELs can be called by mapping the unitigs against a reference genome. By applying the method on 35-fold human resequencing data, we showed that in comparison to the standard pipeline, our approach yields similar accuracy for SNP calling and better results for INDEL calling. It has higher sensitivity than other de novo assembly based methods for variant calling. Our work suggests that variant calling with de novo assembly be a beneficial complement to the standard variant calling pipeline for whole-genome resequencing. In the methodological aspects, we proposed FMD-index for forward-backward extension of DNA sequences, a fast algorithm for finding all super-maximal exact matches and one-pass construction of unitigs from an FMD-index. Availability: http://github.com/lh3/fermi Contact: [email protected]: Rev2: submitted version with minor improvements; 7 page

Hierarchical interpolative factorization for elliptic operators: differential equations

This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF-DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL decomposition that facilitates the efficient inversion of the discretized operator. HIF-DE is based on the multifrontal method but uses skeletonization on the separator fronts to sparsify the dense frontal matrices and thus reduce the cost. We conjecture that this strategy yields linear complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity in 3D can be achieved by skeletonizing the compressed fronts themselves, which amounts geometrically to a recursive dimensional reduction scheme. Numerical experiments support our claims and further demonstrate the performance of our algorithm as a fast direct solver and preconditioner. MATLAB codes are freely available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math. arXiv admin note: substantial text overlap with arXiv:1307.266

Message and time efficient multi-broadcast schemes

We consider message and time efficient broadcasting and multi-broadcasting in wireless ad-hoc networks, where a subset of nodes, each with a unique rumor, wish to broadcast their rumors to all destinations while minimizing the total number of transmissions and total time until all rumors arrive to their destination. Under centralized settings, we introduce a novel approximation algorithm that provides almost optimal results with respect to the number of transmissions and total time, separately. Later on, we show how to efficiently implement this algorithm under distributed settings, where the nodes have only local information about their surroundings. In addition, we show multiple approximation techniques based on the network collision detection capabilities and explain how to calibrate the algorithms' parameters to produce optimal results for time and messages.Comment: In Proceedings FOMC 2013, arXiv:1310.459