19,853 research outputs found
On Rules and Parameter Free Systems in Bounded Arithmetic
We present model–theoretic techniques to obtain conservation
results for first order bounded arithmetic theories, based on a hierarchical
version of the well known notion of an existentially closed model.Ministerio de Educación y Ciencia MTM2005-0865
Existentially Closed Models and Conservation Results in Bounded Arithmetic
We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories Si2 and Ti2 and prove that they are ∀Σbi conservative over their inference rule counterparts, and ∃∀Σbi conservative over their parameter-free versions. A similar analysis of the Σbi-replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.Ministerio de Educación y Ciencia MTM2005-08658Junta de AndalucÃa TIC-13
Lipschitz Optimisation for Lipschitz Interpolation
Techniques known as Nonlinear Set Membership prediction, Kinky Inference or
Lipschitz Interpolation are fast and numerically robust approaches to
nonparametric machine learning that have been proposed to be utilised in the
context of system identification and learning-based control. They utilise
presupposed Lipschitz properties in order to compute inferences over unobserved
function values. Unfortunately, most of these approaches rely on exact
knowledge about the input space metric as well as about the Lipschitz constant.
Furthermore, existing techniques to estimate the Lipschitz constants from the
data are not robust to noise or seem to be ad-hoc and typically are decoupled
from the ultimate learning and prediction task. To overcome these limitations,
we propose an approach for optimising parameters of the presupposed metrics by
minimising validation set prediction errors. To avoid poor performance due to
local minima, we propose to utilise Lipschitz properties of the optimisation
objective to ensure global optimisation success. The resulting approach is a
new flexible method for nonparametric black-box learning. We provide
experimental evidence of the competitiveness of our approach on artificial as
well as on real data
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