108,182 research outputs found
Deciding the dimension of effective dimension reduction space for functional and high-dimensional data
In this paper, we consider regression models with a Hilbert-space-valued
predictor and a scalar response, where the response depends on the predictor
only through a finite number of projections. The linear subspace spanned by
these projections is called the effective dimension reduction (EDR) space. To
determine the dimensionality of the EDR space, we focus on the leading
principal component scores of the predictor, and propose two sequential
testing procedures under the assumption that the predictor has an
elliptically contoured distribution. We further extend these procedures and
introduce a test that simultaneously takes into account a large number of
principal component scores. The proposed procedures are supported by theory,
validated by simulation studies, and illustrated by a real-data example. Our
methods and theory are applicable to functional data and high-dimensional
multivariate data.Comment: Published in at http://dx.doi.org/10.1214/10-AOS816 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On an algorithm to decide whether a free group is a free factor of another
We revisit the problem of deciding whether a finitely generated subgroup H is
a free factor of a given free group F. Known algorithms solve this problem in
time polynomial in the sum of the lengths of the generators of H and
exponential in the rank of F. We show that the latter dependency can be made
exponential in the rank difference rank(F) - rank(H), which often makes a
significant change.Comment: 20 page
Continuity of Functional Transducers: A Profinite Study of Rational Functions
A word-to-word function is continuous for a class of languages~
if its inverse maps _languages to~. This notion
provides a basis for an algebraic study of transducers, and was integral to the
characterization of the sequential transducers computable in some circuit
complexity classes.
Here, we report on the decidability of continuity for functional transducers
and some standard classes of regular languages. To this end, we develop a
robust theory rooted in the standard profinite analysis of regular languages.
Since previous algebraic studies of transducers have focused on the sole
structure of the underlying input automaton, we also compare the two algebraic
approaches. We focus on two questions: When are the automaton structure and the
continuity properties related, and when does continuity propagate to
superclasses
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