4,227 research outputs found
Topological invariants from non-restricted quantum groups
We introduce the notion of a relative spherical category. We prove that such
a category gives rise to the generalized Kashaev and Turaev-Viro-type
3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624,
respectively. In this case we show that these invariants are equal and extend
to what we call a relative Homotopy Quantum Field Theory which is a branch of
the Topological Quantum Field Theory founded by E. Witten and M. Atiyah. Our
main examples of relative spherical categories are the categories of finite
dimensional weight modules over non-restricted quantum groups considered by C.
De Concini, V. Kac, C. Procesi, N. Reshetikhin and M. Rosso. These categories
are not semi-simple and have an infinite number of non-isomorphic irreducible
modules all having vanishing quantum dimensions. We also show that these
categories have associated ribbon categories which gives rise to re-normalized
link invariants. In the case of sl(2) these link invariants are the
Alexander-type multivariable invariants defined by Y. Akutsu, T. Deguchi, and
T. Ohtsuki.Comment: 37 pages, 16 figure
Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization'' by V. Koltchinskii
Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and
oracle inequalities in risk minimization'' by V. Koltchinskii [arXiv:0708.0083]Comment: Published at http://dx.doi.org/10.1214/009053606000001073 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On non-asymptotic bounds for estimation in generalized linear models with highly correlated design
We study a high-dimensional generalized linear model and penalized empirical
risk minimization with penalty. Our aim is to provide a non-trivial
illustration that non-asymptotic bounds for the estimator can be obtained
without relying on the chaining technique and/or the peeling device.Comment: Published at http://dx.doi.org/10.1214/074921707000000319 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
-confidence sets in high-dimensional regression
We study a high-dimensional regression model. Aim is to construct a
confidence set for a given group of regression coefficients, treating all other
regression coefficients as nuisance parameters. We apply a one-step procedure
with the square-root Lasso as initial estimator and a multivariate square-root
Lasso for constructing a surrogate Fisher information matrix. The multivariate
square-root Lasso is based on nuclear norm loss with -penalty. We show
that this procedure leads to an asymptotically -distributed pivot, with
a remainder term depending only on the -error of the initial estimator.
We show that under -sparsity conditions on the regression coefficients
the square-root Lasso produces to a consistent estimator of the noise
variance and we establish sharp oracle inequalities which show that the
remainder term is small under further sparsity conditions on and
compatibility conditions on the design.Comment: 22 page
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