128,083 research outputs found
Clustering above Exponential Families with Tempered Exponential Measures
The link with exponential families has allowed -means clustering to be
generalized to a wide variety of data generating distributions in exponential
families and clustering distortions among Bregman divergences. Getting the
framework to work above exponential families is important to lift roadblocks
like the lack of robustness of some population minimizers carved in their
axiomatization. Current generalisations of exponential families like
-exponential families or even deformed exponential families fail at
achieving the goal. In this paper, we provide a new attempt at getting the
complete framework, grounded in a new generalisation of exponential families
that we introduce, tempered exponential measures (TEM). TEMs keep the maximum
entropy axiomatization framework of -exponential families, but instead of
normalizing the measure, normalize a dual called a co-distribution. Numerous
interesting properties arise for clustering such as improved and controllable
robustness for population minimizers, that keep a simple analytic form
Irreducible representations of deformed oscillator algebra and q-special functions
Different generators of a deformed oscillator algebra give rise to
one-parameter families of -exponential functions and -Hermite polynomials
related by generating functions. Connections of the Stieltjes and Hamburger
classical moment problems with the corresponding resolution of unity for the
-coherent states and with 'coordinate' operators - Jacobi matrices, are also
pointed out.Comment: Contribution to the workshop IWCQIS-96 (JINR, Dubna
Unified model for network dynamics exhibiting nonextensive statistics
We introduce a dynamical network model which unifies a number of network
families which are individually known to exhibit -exponential degree
distributions. The present model dynamics incorporates static (non-growing)
self-organizing networks, preferentially growing networks, and (preferentially)
rewiring networks. Further, it exhibits a natural random graph limit. The
proposed model generalizes network dynamics to rewiring and growth modes which
depend on internal topology as well as on a metric imposed by the space they
are embedded in. In all of the networks emerging from the presented model we
find q-exponential degree distributions over a large parameter space. We
comment on the parameter dependence of the corresponding entropic index q for
the degree distributions, and on the behavior of the clustering coefficients
and neighboring connectivity distributions.Comment: 11 pages 8 fig
Models of q-algebra representations: Matrix elements of the q-oscillator algebra
This article continues a study of function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. Here a q analog of the oscillator algebra (not a quantum algebra) is considered. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ``group operators'' on these representation spaces are computed. This ``local'' approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. It is shown that the matrix elements themselves transform irreducibly under the action of the algebra. q analogs of a formula are found for the product of two hypergeometric functions 1F1 and the product of a 1F1 and a Bessel function. They are interpreted here as expansions of the matrix elements of a ``group operator'' (via the exponential mapping) in a tensor product basis (for the tensor product of two irreducible oscillator algebra representations) in terms of the matrix elements in a reduced basis. As a by-product of this analysis an interesting new orthonormal basis was found for a q analog of the BargmannâSegal Hilbert space of entire functions
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