368 research outputs found
Invariances of random fields paths, with applications in Gaussian Process Regression
We study pathwise invariances of centred random fields that can be controlled
through the covariance. A result involving composition operators is obtained in
second-order settings, and we show that various path properties including
additivity boil down to invariances of the covariance kernel. These results are
extended to a broader class of operators in the Gaussian case, via the Lo\`eve
isometry. Several covariance-driven pathwise invariances are illustrated,
including fields with symmetric paths, centred paths, harmonic paths, or sparse
paths. The proposed approach delivers a number of promising results and
perspectives in Gaussian process regression
Understanding Slow Feature Analysis: A Mathematical Framework
Slow feature analysis is an algorithm for unsupervised learning of invariant representations from data with temporal correlations. Here, we present a mathematical analysis of slow feature analysis for the case where the input-output functions are not restricted in complexity. We show that the optimal functions obey a partial differential eigenvalue problem of a type that is common in theoretical physics. This analogy allows the transfer of mathematical techniques and intuitions from physics to concrete applications of slow feature analysis, thereby providing the means for analytical predictions and a better understanding of simulation results. We put particular emphasis on the situation where the input data are generated from a set of statistically independent sources.\ud
The dependence of the optimal functions on the sources is calculated analytically for the cases where the sources have Gaussian or uniform distribution
Antibracket, Antifields and Gauge-Theory Quantization
The antibracket formalism for gauge theories, at both the classical and
quantum level, is reviewed. Gauge transformations and the associated gauge
structure are analyzed in detail. The basic concepts involved in the
antibracket formalism are elucidated. Gauge-fixing, quantum effects, and
anomalies within the field-antifield formalism are developed. The concepts,
issues and constructions are illustrated using eight gauge-theory models.Comment: 191 pages in three files which must be put together, in Latex, to
appear in Physics Report
Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation
We demonstrate how time-integration of stochastic differential equations
(i.e. Brownian dynamics simulations) can be combined with continuum numerical
bifurcation analysis techniques to analyze the dynamics of liquid crystalline
polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the
approach analyzes the (unavailable in closed form) coarse macroscopic
equations, estimating the necessary quantities through appropriately
initialized, short bursts of Brownian dynamics simulation. Through this
approach, both stable and unstable branches of the equilibrium bifurcation
diagram are obtained for the Doi model of LCPs and their coarse stability is
estimated. Additional macroscopic computational tasks enabled through this
approach, such as coarse projective integration and coarse stabilizing
controller design, are also demonstrated
On a model of visual cortex: learning invariance and selectivity
In this paper we present a class of algorithms for similarity learning on spaces of images. The general framework that we introduce is motivated by some well-known hierarchical pre-processing architectures for object recognition which have been developed during the last decade, and which have been in some cases inspired by functional models of the ventral stream of the visual cortex. These architectures are characterized by the construction of a hierarchy of âlocalâ feature representations of the visual stimulus. We show that our framework includes some well-known techniques, and that it is suitable for the analysis of dynamic visual stimuli, presenting a quantitative error analysis in this setting
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