665 research outputs found
Symbolic integration with respect to the Haar measure on the unitary group
We present IntU package for Mathematica computer algebra system. The
presented package performs a symbolic integration of polynomial functions over
the unitary group with respect to unique normalized Haar measure. We describe a
number of special cases which can be used to optimize the calculation speed for
some classes of integrals. We also provide some examples of usage of the
presented package.Comment: 7 pages, two columns, published version, software available at:
https://github.com/iitis/Int
Random decompositions of Eulerian statistics
This paper develops methods to study the distribution of Eulerian statistics
defined by second-order recurrence relations. We define a random process to
decompose the statistics over compositions of integers. It is shown that the
numbers of descents in random involutions and in random derangements are
asymptotically normal with a rate of convergence of order and
respectively.Comment: 28 page
Semidefinite programming, harmonic analysis and coding theory
These lecture notes where presented as a course of the CIMPA summer school in
Manila, July 20-30, 2009, Semidefinite programming in algebraic combinatorics.
This version is an update June 2010
Support and density of the limit -ary search trees distribution
The space requirements of an -ary search tree satisfies a well-known phase
transition: when , the second order asymptotics is Gaussian. When
, it is not Gaussian any longer and a limit of a complex-valued
martingale arises. We show that the distribution of has a square integrable
density on the complex plane, that its support is the whole complex plane, and
that it has finite exponential moments. The proofs are based on the study of
the distributional equation W\egalLoi\sum_{k=1}^mV_k^{\lambda}W_k, where
are the spacings of independent random variables
uniformly distributed on , are independent copies of W
which are also independent of and is a complex
number
Moments of ideal class counting functions
We consider the counting function of ideals in a given ideal class of a
number field of degree . This describes, at least conjecturally, the Fourier
coefficients of an automorphic form on , typically not a Hecke
eigenform and not cuspidal. We compute its moments, and also investigate the
moments of the corresponding cuspidal projection
Convolution formula for the sums of generalized Dirichlet L-functions
Using the Kuznetsov trace formula, we prove a spectral decomposition for the
sums of generalized Dirichlet -functions. Among applications are an explicit
formula relating norms of prime geodesics to moments of symmetric square
-functions and an asymptotic expansion for the average of central values of
generalized Dirichlet -functions.Comment: to appear in Revista Matem\'atica Iberoamerican
Variational Inference in Nonconjugate Models
Mean-field variational methods are widely used for approximate posterior
inference in many probabilistic models. In a typical application, mean-field
methods approximately compute the posterior with a coordinate-ascent
optimization algorithm. When the model is conditionally conjugate, the
coordinate updates are easily derived and in closed form. However, many models
of interest---like the correlated topic model and Bayesian logistic
regression---are nonconjuate. In these models, mean-field methods cannot be
directly applied and practitioners have had to develop variational algorithms
on a case-by-case basis. In this paper, we develop two generic methods for
nonconjugate models, Laplace variational inference and delta method variational
inference. Our methods have several advantages: they allow for easily derived
variational algorithms with a wide class of nonconjugate models; they extend
and unify some of the existing algorithms that have been derived for specific
models; and they work well on real-world datasets. We studied our methods on
the correlated topic model, Bayesian logistic regression, and hierarchical
Bayesian logistic regression
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