70,984 research outputs found
From Simple to Complex and Ultra-complex Systems:\ud A Paradigm Shift Towards Non-Abelian Systems Dynamics
Atoms, molecules, organisms distinguish layers of reality because of the causal links that govern their behavior, both horizontally (atom-atom, molecule-molecule, organism-organism) and vertically (atom-molecule-organism). This is the first intuition of the theory of levels. Even if the further development of the theory will require imposing a number of qualifications to this initial intuition, the idea of a series of entities organized on different levels of complexity will prove correct. Living systems as well as social systems and the human mind present features remarkably different from those characterizing non-living, simple physical and chemical systems. We propose that super-complexity requires at least four different categorical frameworks, provided by the theories of levels of reality, chronotopoids, (generalized) interactions, and anticipation
From Simple to Complex and Ultra-complex Systems:\ud A Paradigm Shift Towards Non-Abelian Systems Dynamics
Atoms, molecules, organisms distinguish layers of reality because of the causal links that govern their behavior, both horizontally (atom-atom, molecule-molecule, organism-organism) and vertically (atom-molecule-organism). This is the first intuition of the theory of levels. Even if the further development of the theory will require imposing a number of qualifications to this initial intuition, the idea of a series of entities organized on different levels of complexity will prove correct. Living systems as well as social systems and the human mind present features remarkably different from those characterizing non-living, simple physical and chemical systems. We propose that super-complexity requires at least four different categorical frameworks, provided by the theories of levels of reality, chronotopoids, (generalized) interactions, and anticipation
Algebraic shortcuts for leave-one-out cross-validation in supervised network inference
Supervised machine learning techniques have traditionally been very successful at reconstructing biological networks, such as protein-ligand interaction, protein-protein interaction and gene regulatory networks. Many supervised techniques for network prediction use linear models on a possibly nonlinear pairwise feature representation of edges. Recently, much emphasis has been placed on the correct evaluation of such supervised models. It is vital to distinguish between using a model to either predict new interactions in a given network or to predict interactions for a new vertex not present in the original network. This distinction matters because (i) the performance might dramatically differ between the prediction settings and (ii) tuning the model hyperparameters to obtain the best possible model depends on the setting of interest. Specific cross-validation schemes need to be used to assess the performance in such different prediction settings. In this work we discuss a state-of-the-art kernel-based network inference technique called two-step kernel ridge regression. We show that this regression model can be trained efficiently, with a time complexity scaling with the number of vertices rather than the number of edges. Furthermore, this framework leads to a series of cross-validation shortcuts that allow one to rapidly estimate the model performance for any relevant network prediction setting. This allows computational biologists to fully assess the capabilities of their models
The Bond-Algebraic Approach to Dualities
An algebraic theory of dualities is developed based on the notion of bond
algebras. It deals with classical and quantum dualities in a unified fashion
explaining the precise connection between quantum dualities and the low
temperature (strong-coupling)/high temperature (weak-coupling) dualities of
classical statistical mechanics (or (Euclidean) path integrals). Its range of
applications includes discrete lattice, continuum field, and gauge theories.
Dualities are revealed to be local, structure-preserving mappings between
model-specific bond algebras that can be implemented as unitary
transformations, or partial isometries if gauge symmetries are involved. This
characterization permits to search systematically for dualities and
self-dualities in quantum models of arbitrary system size, dimensionality and
complexity, and any classical model admitting a transfer matrix representation.
Dualities like exact dimensional reduction, emergent, and gauge-reducing
dualities that solve gauge constraints can be easily understood in terms of
mappings of bond algebras. As a new example, we show that the (\mathbb{Z}_2)
Higgs model is dual to the extended toric code model {\it in any number of
dimensions}. Non-local dual variables and Jordan-Wigner dictionaries are
derived from the local mappings of bond algebras. Our bond-algebraic approach
goes beyond the standard approach to classical dualities, and could help
resolve the long standing problem of obtaining duality transformations for
lattice non-Abelian models. As an illustration, we present new dualities in any
spatial dimension for the quantum Heisenberg model. Finally, we discuss various
applications including location of phase boundaries, spectral behavior and,
notably, we show how bond-algebraic dualities help constrain and realize
fermionization in an arbitrary number of spatial dimensions.Comment: 131 pages, 22 figures. Submitted to Advances in Physics. Second
version including a new section on the eight-vertex model and the correction
of several typo
Quantum Field Theory: Where We Are
We comment on the present status, the concepts and their limitations, and the
successes and open problems of the various approaches to a relativistic quantum
theory of elementary particles, with a hindsight to questions concerning
quantum gravity and string theory.Comment: To appear in: An Assessment of Current Paradigms in the Physics of
Fundamental Phenomena, to be published by Springer Verlag (2006
A Comparative Study of Pairwise Learning Methods based on Kernel Ridge Regression
Many machine learning problems can be formulated as predicting labels for a
pair of objects. Problems of that kind are often referred to as pairwise
learning, dyadic prediction or network inference problems. During the last
decade kernel methods have played a dominant role in pairwise learning. They
still obtain a state-of-the-art predictive performance, but a theoretical
analysis of their behavior has been underexplored in the machine learning
literature.
In this work we review and unify existing kernel-based algorithms that are
commonly used in different pairwise learning settings, ranging from matrix
filtering to zero-shot learning. To this end, we focus on closed-form efficient
instantiations of Kronecker kernel ridge regression. We show that independent
task kernel ridge regression, two-step kernel ridge regression and a linear
matrix filter arise naturally as a special case of Kronecker kernel ridge
regression, implying that all these methods implicitly minimize a squared loss.
In addition, we analyze universality, consistency and spectral filtering
properties. Our theoretical results provide valuable insights in assessing the
advantages and limitations of existing pairwise learning methods.Comment: arXiv admin note: text overlap with arXiv:1606.0427
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