3,001 research outputs found
A note on the axioms for Zilber's pseudo-exponential fields
We show that Zilber's conjecture that complex exponentiation is isomorphic to
his pseudo-exponentiation follows from the a priori simpler conjecture that
they are elementarily equivalent. An analysis of the first-order types in
pseudo-exponentiation leads to a description of the elementary embeddings, and
the result that pseudo-exponential fields are precisely the models of their
common first-order theory which are atomic over exponential transcendence
bases. We also show that the class of all pseudo-exponential fields is an
example of a non-finitary abstract elementary class, answering a question of
Kes\"al\"a and Baldwin.Comment: 10 pages, v2: substantial alteration
MDL Convergence Speed for Bernoulli Sequences
The Minimum Description Length principle for online sequence
estimation/prediction in a proper learning setup is studied. If the underlying
model class is discrete, then the total expected square loss is a particularly
interesting performance measure: (a) this quantity is finitely bounded,
implying convergence with probability one, and (b) it additionally specifies
the convergence speed. For MDL, in general one can only have loss bounds which
are finite but exponentially larger than those for Bayes mixtures. We show that
this is even the case if the model class contains only Bernoulli distributions.
We derive a new upper bound on the prediction error for countable Bernoulli
classes. This implies a small bound (comparable to the one for Bayes mixtures)
for certain important model classes. We discuss the application to Machine
Learning tasks such as classification and hypothesis testing, and
generalization to countable classes of i.i.d. models.Comment: 28 page
Similarity and Coincidence Isometries for Modules
The groups of (linear) similarity and coincidence isometries of certain
modules in d-dimensional Euclidean space, which naturally occur in
quasicrystallography, are considered. It is shown that the structure of the
factor group of similarity modulo coincidence isometries is the direct sum of
cyclic groups of prime power orders that divide d. In particular, if the
dimension d is a prime number p, the factor group is an elementary Abelian
p-group. This generalizes previous results obtained for lattices to situations
relevant in quasicrystallography.Comment: 14 page
Cross sections for geodesic flows and \alpha-continued fractions
We adjust Arnoux's coding, in terms of regular continued fractions, of the
geodesic flow on the modular surface to give a cross section on which the
return map is a double cover of the natural extension for the \alpha-continued
fractions, for each in (0,1]. The argument is sufficiently robust to
apply to the Rosen continued fractions and their recently introduced
\alpha-variants.Comment: 20 pages, 2 figure
On the Convergence Speed of MDL Predictions for Bernoulli Sequences
We consider the Minimum Description Length principle for online sequence
prediction. If the underlying model class is discrete, then the total expected
square loss is a particularly interesting performance measure: (a) this
quantity is bounded, implying convergence with probability one, and (b) it
additionally specifies a `rate of convergence'. Generally, for MDL only
exponential loss bounds hold, as opposed to the linear bounds for a Bayes
mixture. We show that this is even the case if the model class contains only
Bernoulli distributions. We derive a new upper bound on the prediction error
for countable Bernoulli classes. This implies a small bound (comparable to the
one for Bayes mixtures) for certain important model classes. The results apply
to many Machine Learning tasks including classification and hypothesis testing.
We provide arguments that our theorems generalize to countable classes of
i.i.d. models.Comment: 17 page
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