60,614 research outputs found
Sparse Linear Identifiable Multivariate Modeling
In this paper we consider sparse and identifiable linear latent variable
(factor) and linear Bayesian network models for parsimonious analysis of
multivariate data. We propose a computationally efficient method for joint
parameter and model inference, and model comparison. It consists of a fully
Bayesian hierarchy for sparse models using slab and spike priors (two-component
delta-function and continuous mixtures), non-Gaussian latent factors and a
stochastic search over the ordering of the variables. The framework, which we
call SLIM (Sparse Linear Identifiable Multivariate modeling), is validated and
bench-marked on artificial and real biological data sets. SLIM is closest in
spirit to LiNGAM (Shimizu et al., 2006), but differs substantially in
inference, Bayesian network structure learning and model comparison.
Experimentally, SLIM performs equally well or better than LiNGAM with
comparable computational complexity. We attribute this mainly to the stochastic
search strategy used, and to parsimony (sparsity and identifiability), which is
an explicit part of the model. We propose two extensions to the basic i.i.d.
linear framework: non-linear dependence on observed variables, called SNIM
(Sparse Non-linear Identifiable Multivariate modeling) and allowing for
correlations between latent variables, called CSLIM (Correlated SLIM), for the
temporal and/or spatial data. The source code and scripts are available from
http://cogsys.imm.dtu.dk/slim/.Comment: 45 pages, 17 figure
Grothendieck's theorem on non-abelian H^2 and local-global principles
A theorem of Grothendieck asserts that over a perfect field k of
cohomological dimension one, all non-abelian H^2-cohomology sets of algebraic
groups are trivial. The purpose of this paper is to establish a formally real
generalization of this theorem. The generalization -- to the context of perfect
fields of virtual cohomological dimension one -- takes the form of a
local-global principle for the H^2-sets with respect to the orderings of the
field. This principle asserts in particular that an element in H^2 is neutral
precisely when it is neutral in the real closure with respect to every ordering
in a dense subset of the real spectrum of k. Our techniques provide a new proof
of Grothendieck's original theorem. An application to homogeneous spaces over k
is also given.Comment: 22 pages, AMS-TeX; accepted for publication by the Journal of the AM
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
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