822 research outputs found
Robust and Sparse Regression via -divergence
In high-dimensional data, many sparse regression methods have been proposed.
However, they may not be robust against outliers. Recently, the use of density
power weight has been studied for robust parameter estimation and the
corresponding divergences have been discussed. One of such divergences is the
-divergence and the robust estimator using the -divergence is
known for having a strong robustness. In this paper, we consider the robust and
sparse regression based on -divergence. We extend the
-divergence to the regression problem and show that it has a strong
robustness under heavy contamination even when outliers are heterogeneous. The
loss function is constructed by an empirical estimate of the
-divergence with sparse regularization and the parameter estimate is
defined as the minimizer of the loss function. To obtain the robust and sparse
estimate, we propose an efficient update algorithm which has a monotone
decreasing property of the loss function. Particularly, we discuss a linear
regression problem with regularization in detail. In numerical
experiments and real data analyses, we see that the proposed method outperforms
past robust and sparse methods.Comment: 25 page
Postquantum Br\`{e}gman relative entropies and nonlinear resource theories
We introduce the family of postquantum Br\`{e}gman relative entropies, based
on nonlinear embeddings into reflexive Banach spaces (with examples given by
reflexive noncommutative Orlicz spaces over semi-finite W*-algebras,
nonassociative L spaces over semi-finite JBW-algebras, and noncommutative
L spaces over arbitrary W*-algebras). This allows us to define a class of
geometric categories for nonlinear postquantum inference theory (providing an
extension of Chencov's approach to foundations of statistical inference), with
constrained maximisations of Br\`{e}gman relative entropies as morphisms and
nonlinear images of closed convex sets as objects. Further generalisation to a
framework for nonlinear convex operational theories is developed using a larger
class of morphisms, determined by Br\`{e}gman nonexpansive operations (which
provide a well-behaved family of Mielnik's nonlinear transmitters). As an
application, we derive a range of nonlinear postquantum resource theories
determined in terms of this class of operations.Comment: v2: several corrections and improvements, including an extension to
the postquantum (generally) and JBW-algebraic (specifically) cases, a section
on nonlinear resource theories, and more informative paper's titl
Extension of information geometry for modelling non-statistical systems
In this dissertation, an abstract formalism extending information geometry is
introduced. This framework encompasses a broad range of modelling problems,
including possible applications in machine learning and in the information
theoretical foundations of quantum theory. Its purely geometrical foundations
make no use of probability theory and very little assumptions about the data or
the models are made. Starting only from a divergence function, a Riemannian
geometrical structure consisting of a metric tensor and an affine connection is
constructed and its properties are investigated. Also the relation to
information geometry and in particular the geometry of exponential families of
probability distributions is elucidated. It turns out this geometrical
framework offers a straightforward way to determine whether or not a
parametrised family of distributions can be written in exponential form. Apart
from the main theoretical chapter, the dissertation also contains a chapter of
examples illustrating the application of the formalism and its geometric
properties, a brief introduction to differential geometry and a historical
overview of the development of information geometry.Comment: PhD thesis, University of Antwerp, Advisors: Prof. dr. Jan Naudts and
Prof. dr. Jacques Tempere, December 2014, 108 page
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