1,384 research outputs found
Tensor Analysis and Fusion of Multimodal Brain Images
Current high-throughput data acquisition technologies probe dynamical systems
with different imaging modalities, generating massive data sets at different
spatial and temporal resolutions posing challenging problems in multimodal data
fusion. A case in point is the attempt to parse out the brain structures and
networks that underpin human cognitive processes by analysis of different
neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the
multimodal, multi-scale nature of neuroimaging data is well reflected by a
multi-way (tensor) structure where the underlying processes can be summarized
by a relatively small number of components or "atoms". We introduce
Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network
notation in order to analyze these models. These diagrams not only clarify
matrix and tensor EEG and fMRI time/frequency analysis and inverse problems,
but also help understand multimodal fusion via Multiway Partial Least Squares
and Coupled Matrix-Tensor Factorization. We show here, for the first time, that
Granger causal analysis of brain networks is a tensor regression problem, thus
allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI
recordings shows the potential of the methods and suggests their use in other
scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE
The mirror conjecture for minuscule flag varieties
We prove Rietsch's mirror conjecture that the Dubrovin quantum connection for
minuscule flag varieties is isomorphic to the character D-module of the
Berenstein-Kazhdan geometric crystal. The idea is to recognize the quantum
connection as Galois and the geometric crystal as automorphic. We reveal
surprising relations with the works of Frenkel-Gross, Heinloth-Ng\^o-Yun and
Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results
where Hecke eigensheaves are determined by their local ramification. As
corollaries we obtain combinatorial identities for counts of rational curves
and the Peterson variety presentation of the small quantum cohomology ring
Laguerre and Meixner orthogonal bases in the algebra of symmetric functions
Analogs of Laguerre and Meixner orthogonal polynomials in the algebra of
symmetric functions are studied. This is a detailed exposition of part of the
results announced in arXiv:1009.2037. The work is motivated by a connection
with a model of infinite-dimensional Markov dynamics.Comment: Latex, 52p
Joining Extractions of Regular Expressions
Regular expressions with capture variables, also known as "regex formulas,"
extract relations of spans (interval positions) from text. These relations can
be further manipulated via Relational Algebra as studied in the context of
document spanners, Fagin et al.'s formal framework for information extraction.
We investigate the complexity of querying text by Conjunctive Queries (CQs) and
Unions of CQs (UCQs) on top of regex formulas. We show that the lower bounds
(NP-completeness and W[1]-hardness) from the relational world also hold in our
setting; in particular, hardness hits already single-character text! Yet, the
upper bounds from the relational world do not carry over. Unlike the relational
world, acyclic CQs, and even gamma-acyclic CQs, are hard to compute. The source
of hardness is that it may be intractable to instantiate the relation defined
by a regex formula, simply because it has an exponential number of tuples. Yet,
we are able to establish general upper bounds. In particular, UCQs can be
evaluated with polynomial delay, provided that every CQ has a bounded number of
atoms (while unions and projection can be arbitrary). Furthermore, UCQ
evaluation is solvable with FPT (Fixed-Parameter Tractable) delay when the
parameter is the size of the UCQ
Model selection in the space of Gaussian models invariant by symmetry
We consider multivariate centred Gaussian models for the random variable
, invariant under the action of a subgroup of the group of
permutations on . Using the representation theory of the
symmetric group on the field of reals, we derive the distribution of the
maximum likelihood estimate of the covariance parameter and also the
analytic expression of the normalizing constant of the Diaconis-Ylvisaker
conjugate prior for the precision parameter . We can thus
perform Bayesian model selection in the class of complete Gaussian models
invariant by the action of a subgroup of the symmetric group, which we could
also call complete RCOP models. We illustrate our results with a toy example of
dimension and several examples for selection within cyclic groups,
including a high dimensional example with .Comment: 34 pages, 4 figures, 5 table
Computable de Finetti measures
We prove a computable version of de Finetti's theorem on exchangeable
sequences of real random variables. As a consequence, exchangeable stochastic
processes expressed in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor
corrections. To appear in Annals of Pure and Applied Logic. Extended abstract
appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23
FeynGKZ: a Mathematica package for solving Feynman integrals using GKZ hypergeometric systems
In the Lee-Pomeransky representation, Feynman integrals can be identified as
a subset of Euler-Mellin integrals, which are known to satisfy
Gel'fand-Kapranov-Zelevinsky (GKZ) system of partial differential equations.
Here we present an automated package to derive the associated GKZ system for a
given Feynman diagram and solve it in terms of hypergeometric functions using
two equivalent algorithms, namely the triangulation method and the Gr\"obner
deformation method. We present our code in the form of a Mathematica package
FeynGKZ.wl which requires the softwares polymake, Macaulay2 and TOPCOM, and the
packages AMBRE and Olsson.wl as dependencies. As applications of the package,
we find series solutions to the GKZ systems of several one-loop and two-loop
Feynman integrals. These are included in the file Examples.nb that can be
downloaded along with the package from https://github.com/anant-group/FeynGKZ.Comment: 26 pages, 1 figure, code repository:
https://github.com/anant-group/FeynGK
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