1,755 research outputs found
Predictability, complexity and learning
We define {\em predictive information} as the mutual
information between the past and the future of a time series. Three
qualitatively different behaviors are found in the limit of large observation
times : can remain finite, grow logarithmically, or grow
as a fractional power law. If the time series allows us to learn a model with a
finite number of parameters, then grows logarithmically with
a coefficient that counts the dimensionality of the model space. In contrast,
power--law growth is associated, for example, with the learning of infinite
parameter (or nonparametric) models such as continuous functions with
smoothness constraints. There are connections between the predictive
information and measures of complexity that have been defined both in learning
theory and in the analysis of physical systems through statistical mechanics
and dynamical systems theory. Further, in the same way that entropy provides
the unique measure of available information consistent with some simple and
plausible conditions, we argue that the divergent part of
provides the unique measure for the complexity of dynamics underlying a time
series. Finally, we discuss how these ideas may be useful in different problems
in physics, statistics, and biology.Comment: 53 pages, 3 figures, 98 references, LaTeX2
The Shape of Learning Curves: a Review
Learning curves provide insight into the dependence of a learner's
generalization performance on the training set size. This important tool can be
used for model selection, to predict the effect of more training data, and to
reduce the computational complexity of model training and hyperparameter
tuning. This review recounts the origins of the term, provides a formal
definition of the learning curve, and briefly covers basics such as its
estimation. Our main contribution is a comprehensive overview of the literature
regarding the shape of learning curves. We discuss empirical and theoretical
evidence that supports well-behaved curves that often have the shape of a power
law or an exponential. We consider the learning curves of Gaussian processes,
the complex shapes they can display, and the factors influencing them. We draw
specific attention to examples of learning curves that are ill-behaved, showing
worse learning performance with more training data. To wrap up, we point out
various open problems that warrant deeper empirical and theoretical
investigation. All in all, our review underscores that learning curves are
surprisingly diverse and no universal model can be identified
Understanding and Controlling Regime Switching in Molecular Diffusion
Diffusion can be strongly affected by ballistic flights (long jumps) as well
as long-lived sticking trajectories (long sticks). Using statistical inference
techniques in the spirit of Granger causality, we investigate the appearance of
long jumps and sticks in molecular-dynamics simulations of diffusion in a
prototype system, a benzene molecule on a graphite substrate. We find that
specific fluctuations in certain, but not all, internal degrees of freedom of
the molecule can be linked to either long jumps or sticks. Furthermore, by
changing the prevalence of these predictors with an outside influence, the
diffusion of the molecule can be controlled. The approach presented in this
proof of concept study is very generic, and can be applied to larger and more
complex molecules. Additionally, the predictor variables can be chosen in a
general way so as to be accessible in experiments, making the method feasible
for control of diffusion in applications. Our results also demonstrate that
data-mining techniques can be used to investigate the phase-space structure of
high-dimensional nonlinear dynamical systems.Comment: accepted for publication by PR
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