47,208 research outputs found
Invariant Tensors Formulae via Chord Diagrams
We provide an explicit algorithm to calculate invariant tensors for the
adjoint representation of the simple Lie algebra , as well as arbitrary
representation in terms of roots. We also obtain explicit formulae for the
adjoint representations of the orthogonal and symplectic Lie algebras
and .Comment: 18 pages, 8 figures. To appear in a special issue of Journal of
Mathematical Science
The Mode of Computing
The Turing Machine is the paradigmatic case of computing machines, but there
are others, such as Artificial Neural Networks, Table Computing,
Relational-Indeterminate Computing and diverse forms of analogical computing,
each of which based on a particular underlying intuition of the phenomenon of
computing. This variety can be captured in terms of system levels,
re-interpreting and generalizing Newell's hierarchy, which includes the
knowledge level at the top and the symbol level immediately below it. In this
re-interpretation the knowledge level consists of human knowledge and the
symbol level is generalized into a new level that here is called The Mode of
Computing. Natural computing performed by the brains of humans and non-human
animals with a developed enough neural system should be understood in terms of
a hierarchy of system levels too. By analogy from standard computing machinery
there must be a system level above the neural circuitry levels and directly
below the knowledge level that is named here The mode of Natural Computing. A
central question for Cognition is the characterization of this mode. The Mode
of Computing provides a novel perspective on the phenomena of computing,
interpreting, the representational and non-representational views of cognition,
and consciousness.Comment: 35 pages, 8 figure
Applying MDL to Learning Best Model Granularity
The Minimum Description Length (MDL) principle is solidly based on a provably
ideal method of inference using Kolmogorov complexity. We test how the theory
behaves in practice on a general problem in model selection: that of learning
the best model granularity. The performance of a model depends critically on
the granularity, for example the choice of precision of the parameters. Too
high precision generally involves modeling of accidental noise and too low
precision may lead to confusion of models that should be distinguished. This
precision is often determined ad hoc. In MDL the best model is the one that
most compresses a two-part code of the data set: this embodies ``Occam's
Razor.'' In two quite different experimental settings the theoretical value
determined using MDL coincides with the best value found experimentally. In the
first experiment the task is to recognize isolated handwritten characters in
one subject's handwriting, irrespective of size and orientation. Based on a new
modification of elastic matching, using multiple prototypes per character, the
optimal prediction rate is predicted for the learned parameter (length of
sampling interval) considered most likely by MDL, which is shown to coincide
with the best value found experimentally. In the second experiment the task is
to model a robot arm with two degrees of freedom using a three layer
feed-forward neural network where we need to determine the number of nodes in
the hidden layer giving best modeling performance. The optimal model (the one
that extrapolizes best on unseen examples) is predicted for the number of nodes
in the hidden layer considered most likely by MDL, which again is found to
coincide with the best value found experimentally.Comment: LaTeX, 32 pages, 5 figures. Artificial Intelligence journal, To
appea
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