14,501 research outputs found
Building a path-integral calculus: a covariant discretization approach
Path integrals are a central tool when it comes to describing quantum or
thermal fluctuations of particles or fields. Their success dates back to
Feynman who showed how to use them within the framework of quantum mechanics.
Since then, path integrals have pervaded all areas of physics where fluctuation
effects, quantum and/or thermal, are of paramount importance. Their appeal is
based on the fact that one converts a problem formulated in terms of operators
into one of sampling classical paths with a given weight. Path integrals are
the mirror image of our conventional Riemann integrals, with functions
replacing the real numbers one usually sums over. However, unlike conventional
integrals, path integration suffers a serious drawback: in general, one cannot
make non-linear changes of variables without committing an error of some sort.
Thus, no path-integral based calculus is possible. Here we identify which are
the deep mathematical reasons causing this important caveat, and we come up
with cures for systems described by one degree of freedom. Our main result is a
construction of path integration free of this longstanding problem, through a
direct time-discretization procedure.Comment: 22 pages, 2 figures, 1 table. Typos correcte
Encrypted statistical machine learning: new privacy preserving methods
We present two new statistical machine learning methods designed to learn on
fully homomorphic encrypted (FHE) data. The introduction of FHE schemes
following Gentry (2009) opens up the prospect of privacy preserving statistical
machine learning analysis and modelling of encrypted data without compromising
security constraints. We propose tailored algorithms for applying extremely
random forests, involving a new cryptographic stochastic fraction estimator,
and na\"{i}ve Bayes, involving a semi-parametric model for the class decision
boundary, and show how they can be used to learn and predict from encrypted
data. We demonstrate that these techniques perform competitively on a variety
of classification data sets and provide detailed information about the
computational practicalities of these and other FHE methods.Comment: 39 page
The extension problem for partial Boolean structures in Quantum Mechanics
Alternative partial Boolean structures, implicit in the discussion of
classical representability of sets of quantum mechanical predictions, are
characterized, with definite general conclusions on the equivalence of the
approaches going back to Bell and Kochen-Specker. An algebraic approach is
presented, allowing for a discussion of partial classical extension, amounting
to reduction of the number of contexts, classical representability arising as a
special case. As a result, known techniques are generalized and some of the
associated computational difficulties overcome. The implications on the
discussion of Boole-Bell inequalities are indicated.Comment: A number of misprints have been corrected and some terminology
changed in order to avoid possible ambiguitie
Multiplicative local linear hazard estimation and best one-sided cross-validation
This paper develops detailed mathematical statistical theory of a new class of cross-validation techniques of local linear kernel hazards and their multiplicative bias corrections. The new class of cross-validation combines principles of local information and recent advances in indirect cross-validation. A few applications of cross-validating multiplicative kernel hazard estimation do exist in the literature. However, detailed mathematical statistical theory and small sample performance are introduced via this paper and further upgraded to our new class of best one-sided cross-validation. Best one-sided cross-validation turns out to have excellent performance in its practical illustrations, in its small sample performance and in its mathematical statistical theoretical performance
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