1,701 research outputs found
Learning loopy graphical models with latent variables: Efficient methods and guarantees
The problem of structure estimation in graphical models with latent variables
is considered. We characterize conditions for tractable graph estimation and
develop efficient methods with provable guarantees. We consider models where
the underlying Markov graph is locally tree-like, and the model is in the
regime of correlation decay. For the special case of the Ising model, the
number of samples required for structural consistency of our method scales
as , where p is the
number of variables, is the minimum edge potential, is
the depth (i.e., distance from a hidden node to the nearest observed nodes),
and is a parameter which depends on the bounds on node and edge
potentials in the Ising model. Necessary conditions for structural consistency
under any algorithm are derived and our method nearly matches the lower bound
on sample requirements. Further, the proposed method is practical to implement
and provides flexibility to control the number of latent variables and the
cycle lengths in the output graph.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1070 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Effective Field Theory for Few-Nucleon Systems
We review the effective field theories (EFTs) developed for few-nucleon
systems. These EFTs are controlled expansions in momenta, where certain
(leading-order) interactions are summed to all orders. At low energies, an EFT
with only contact interactions allows a detailed analysis of renormalization in
a non-perturbative context and uncovers novel asymptotic behavior. Manifestly
model-independent calculations can be carried out to high orders, leading to
high precision. At higher energies, an EFT that includes pion fields justifies
and extends the traditional framework of phenomenological potentials. The
correct treatment of QCD symmetries ensures a connection with lattice QCD.
Several tests and prospects of these EFTs are discussed.Comment: 55 pages, 18 figures, to appear in Ann. Rev. Nucl. Part. Sci. 52
(2002
Tracing evolutionary links between species
The idea that all life on earth traces back to a common beginning dates back
at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists
have tried to piece together parts of this `tree of life' based on what we can
observe today: fossils, and the evolutionary signal that is present in the
genomes and phenotypes of different organisms. Mathematics has played a key
role in helping transform genetic data into phylogenetic (evolutionary) trees
and networks. Here, I will explain some of the central concepts and basic
results in phylogenetics, which benefit from several branches of mathematics,
including combinatorics, probability and algebra.Comment: 18 pages, 6 figures (Invited review paper (draft version) for AMM
Adinkras From Ordered Quartets of BC Coxeter Group Elements and Regarding 1,358,954,496 Matrix Elements of the Gadget
We examine values of the Adinkra Holoraumy-induced Gadget representation
space metric over all possible four-color, four-open node, and four-closed node
adinkras. Of the 1,358,954,496 gadget matrix elements, only 226,492,416 are
non-vanishing and take on one of three values: , , or and thus a
subspace isomorphic to a description of a body-centered tetrahedral molecule
emerges.Comment: LaTeX twice, 56pp, 30 tables, 5 figures, latest version includes link
to updated code, minor corrections, and additional support about inequivalent
representations and tetrahedral geometry comments added along with
observations about similarity with results previously found by Nekraso
Counting 4-Patterns in Permutations Is Equivalent to Counting 4-Cycles in Graphs
Permutation ? appears in permutation ? if there exists a subsequence of ? that is order-isomorphic to ?. The natural algorithmic question is to check if ? appears in ?, and if so count the number of occurrences. Only since very recently we know that for any fixed length k, we can check if a given pattern of length k appears in a permutation of length n in time linear in n, but being able to count all such occurrences in f(k)? n^o(k/log k) time would refute the exponential time hypothesis (ETH). Together with practical applications in statistics, this motivates a systematic study of the complexity of counting occurrences for different patterns of fixed small length k. We investigate this question for k = 4. Very recently, Even-Zohar and Leng [arXiv 2019] identified two types of 4-patterns. For the first type they designed an ??(n) time algorithm, while for the second they were able to provide an ??(n^1.5) time algorithm. This brings up the question whether the permutations of the second type are inherently harder than the first type.
We establish a connection between counting 4-patterns of the second type and counting 4-cycles (not necessarily induced) in a sparse undirected graph. By designing two-way reductions we show that the complexities of both problems are the same, up to polylogarithmic factors. This allows us to leverage the work done on the latter to provide a reasonable argument for why there is a difference in the complexities for counting 4-patterns of the first and the second type. In particular, even for the seemingly simpler problem of detecting a 4-cycle in a graph on m edges, the best known algorithm works in ?(m^{4/3}) time. Our reductions imply that an ?(n^{4/3-?}) time algorithm for counting occurrences of any 4-pattern of the second type in a permutation of length n would imply an exciting breakthrough for counting (and hence also detecting) 4-cycles. In the other direction, by plugging in the fastest known algorithm for counting 4-cycles, we obtain an algorithm for counting occurrences of any 4-pattern of the second type in ?(n^1.48) time
Generalizations of the genomic rank distance to indels
MOTIVATION: The rank distance model represents genome rearrangements in multi-chromosomal genomes as matrix operations, which allows the reconstruction of parsimonious histories of evolution by rearrangements. We seek to generalize this model by allowing for genomes with different gene content, to accommodate a broader range of biological contexts. We approach this generalization by using a matrix representation of genomes. This leads to simple distance formulas and sorting algorithms for genomes with different gene contents, but without duplications. RESULTS: We generalize the rank distance to genomes with different gene content in two different ways. The first approach adds insertions, deletions and the substitution of a single extremity to the basic operations. We show how to efficiently compute this distance. To avoid genomes with incomplete markers, our alternative distance, the rank-indel distance, only uses insertions and deletions of entire chromosomes. We construct phylogenetic trees with our distances and the DCJ-Indel distance for simulated data and real prokaryotic genomes, and compare them against reference trees. For simulated data, our distances outperform the DCJ-Indel distance using the Quartet metric as baseline. This suggests that rank distances are more robust for comparing distantly related species. For real prokaryotic genomes, all rearrangement-based distances yield phylogenetic trees that are topologically distant from the reference (65% similarity with Quartet metric), but are able to cluster related species within their respective clades and distinguish the Shigella strains as the farthest relative of the Escherichia coli strains, a feature not seen in the reference tree. AVAILABILITY AND IMPLEMENTATION: Code and instructions are available at https://github.com/meidanis-lab/rank-indel. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online
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