21,888 research outputs found
Geometrical complexity of data approximators
There are many methods developed to approximate a cloud of vectors embedded
in high-dimensional space by simpler objects: starting from principal points
and linear manifolds to self-organizing maps, neural gas, elastic maps, various
types of principal curves and principal trees, and so on. For each type of
approximators the measure of the approximator complexity was developed too.
These measures are necessary to find the balance between accuracy and
complexity and to define the optimal approximations of a given type. We propose
a measure of complexity (geometrical complexity) which is applicable to
approximators of several types and which allows comparing data approximations
of different types.Comment: 10 pages, 3 figures, minor correction and extensio
Data complexity measured by principal graphs
How to measure the complexity of a finite set of vectors embedded in a
multidimensional space? This is a non-trivial question which can be approached
in many different ways. Here we suggest a set of data complexity measures using
universal approximators, principal cubic complexes. Principal cubic complexes
generalise the notion of principal manifolds for datasets with non-trivial
topologies. The type of the principal cubic complex is determined by its
dimension and a grammar of elementary graph transformations. The simplest
grammar produces principal trees.
We introduce three natural types of data complexity: 1) geometric (deviation
of the data's approximator from some "idealized" configuration, such as
deviation from harmonicity); 2) structural (how many elements of a principal
graph are needed to approximate the data), and 3) construction complexity (how
many applications of elementary graph transformations are needed to construct
the principal object starting from the simplest one).
We compute these measures for several simulated and real-life data
distributions and show them in the "accuracy-complexity" plots, helping to
optimize the accuracy/complexity ratio. We discuss various issues connected
with measuring data complexity. Software for computing data complexity measures
from principal cubic complexes is provided as well.Comment: Computers and Mathematics with Applications, in pres
The Prediction value
We introduce the prediction value (PV) as a measure of players' informational
importance in probabilistic TU games. The latter combine a standard TU game and
a probability distribution over the set of coalitions. Player 's prediction
value equals the difference between the conditional expectations of when
cooperates or not. We characterize the prediction value as a special member
of the class of (extended) values which satisfy anonymity, linearity and a
consistency property. Every -player binomial semivalue coincides with the PV
for a particular family of probability distributions over coalitions. The PV
can thus be regarded as a power index in specific cases. Conversely, some
semivalues -- including the Banzhaf but not the Shapley value -- can be
interpreted in terms of informational importance.Comment: 26 pages, 2 table
Theoretical framework for quantum networks
We present a framework to treat quantum networks and all possible
transformations thereof, including as special cases all possible manipulations
of quantum states, measurements, and channels, such as, e.g., cloning,
discrimination, estimation, and tomography. Our framework is based on the
concepts of quantum comb-which describes all transformations achievable by a
given quantum network-and link product-the operation of connecting two quantum
networks. Quantum networks are treated both from a constructive point of
view-based on connections of elementary circuits-and from an axiomatic
one-based on a hierarchy of admissible quantum maps. In the axiomatic context a
fundamental property is shown, which we call universality of quantum memory
channels: any admissible transformation of quantum networks can be realized by
a suitable sequence of memory channels. The open problem whether this property
fails for some nonquantum theory, e.g., for no-signaling boxes, is posed.Comment: 23 pages, revtex
Lorenz, G\"{o}del and Penrose: New perspectives on determinism and causality in fundamental physics
Despite being known for his pioneering work on chaotic unpredictability, the
key discovery at the core of meteorologist Ed Lorenz's work is the link between
space-time calculus and state-space fractal geometry. Indeed, properties of
Lorenz's fractal invariant set relate space-time calculus to deep areas of
mathematics such as G\"{o}del's Incompleteness Theorem. These properties,
combined with some recent developments in theoretical and observational
cosmology, motivate what is referred to as the `cosmological invariant set
postulate': that the universe can be considered a deterministic dynamical
system evolving on a causal measure-zero fractal invariant set in its
state space. Symbolic representations of are constructed explicitly based
on permutation representations of quaternions. The resulting `invariant set
theory' provides some new perspectives on determinism and causality in
fundamental physics. For example, whilst the cosmological invariant set appears
to have a rich enough structure to allow a description of quantum probability,
its measure-zero character ensures it is sparse enough to prevent invariant set
theory being constrained by the Bell inequality (consistent with a partial
violation of the so-called measurement independence postulate). The primacy of
geometry as embodied in the proposed theory extends the principles underpinning
general relativity. As a result, the physical basis for contemporary programmes
which apply standard field quantisation to some putative gravitational
lagrangian is questioned. Consistent with Penrose's suggestion of a
deterministic but non-computable theory of fundamental physics, a
`gravitational theory of the quantum' is proposed based on the geometry of
, with potential observational consequences for the dark universe.Comment: This manuscript has been accepted for publication in Contemporary
Physics and is based on the author's 9th Dennis Sciama Lecture, given in
Oxford and Triest
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