4,263 research outputs found
Universal bound on the cardinality of local hidden variables in networks
We present an algebraic description of the sets of local correlations in
arbitrary networks, when the parties have finite inputs and outputs. We
consider networks generalizing the usual Bell scenarios by the presence of
multiple uncorrelated sources. We prove a finite upper bound on the cardinality
of the value sets of the local hidden variables. Consequently, we find that the
sets of local correlations are connected, closed and semialgebraic, and bounded
by tight polynomial Bell-like inequalities.Comment: 5 pages + 6 pages appendix, 2 figures, comments welcom
Hierarchical Models as Marginals of Hierarchical Models
We investigate the representation of hierarchical models in terms of
marginals of other hierarchical models with smaller interactions. We focus on
binary variables and marginals of pairwise interaction models whose hidden
variables are conditionally independent given the visible variables. In this
case the problem is equivalent to the representation of linear subspaces of
polynomials by feedforward neural networks with soft-plus computational units.
We show that every hidden variable can freely model multiple interactions among
the visible variables, which allows us to generalize and improve previous
results. In particular, we show that a restricted Boltzmann machine with less
than hidden binary variables can approximate
every distribution of visible binary variables arbitrarily well, compared
to from the best previously known result.Comment: 18 pages, 4 figures, 2 tables, WUPES'1
A Theory of Cheap Control in Embodied Systems
We present a framework for designing cheap control architectures for embodied
agents. Our derivation is guided by the classical problem of universal
approximation, whereby we explore the possibility of exploiting the agent's
embodiment for a new and more efficient universal approximation of behaviors
generated by sensorimotor control. This embodied universal approximation is
compared with the classical non-embodied universal approximation. To exemplify
our approach, we present a detailed quantitative case study for policy models
defined in terms of conditional restricted Boltzmann machines. In contrast to
non-embodied universal approximation, which requires an exponential number of
parameters, in the embodied setting we are able to generate all possible
behaviors with a drastically smaller model, thus obtaining cheap universal
approximation. We test and corroborate the theory experimentally with a
six-legged walking machine. The experiments show that the sufficient controller
complexity predicted by our theory is tight, which means that the theory has
direct practical implications. Keywords: cheap design, embodiment, sensorimotor
loop, universal approximation, conditional restricted Boltzmann machineComment: 27 pages, 10 figure
Universal Approximation Depth and Errors of Narrow Belief Networks with Discrete Units
We generalize recent theoretical work on the minimal number of layers of
narrow deep belief networks that can approximate any probability distribution
on the states of their visible units arbitrarily well. We relax the setting of
binary units (Sutskever and Hinton, 2008; Le Roux and Bengio, 2008, 2010;
Mont\'ufar and Ay, 2011) to units with arbitrary finite state spaces, and the
vanishing approximation error to an arbitrary approximation error tolerance.
For example, we show that a -ary deep belief network with layers of width for some can approximate any probability
distribution on without exceeding a Kullback-Leibler
divergence of . Our analysis covers discrete restricted Boltzmann
machines and na\"ive Bayes models as special cases.Comment: 19 pages, 5 figures, 1 tabl
When Does a Mixture of Products Contain a Product of Mixtures?
We derive relations between theoretical properties of restricted Boltzmann
machines (RBMs), popular machine learning models which form the building blocks
of deep learning models, and several natural notions from discrete mathematics
and convex geometry. We give implications and equivalences relating
RBM-representable probability distributions, perfectly reconstructible inputs,
Hamming modes, zonotopes and zonosets, point configurations in hyperplane
arrangements, linear threshold codes, and multi-covering numbers of hypercubes.
As a motivating application, we prove results on the relative representational
power of mixtures of product distributions and products of mixtures of pairs of
product distributions (RBMs) that formally justify widely held intuitions about
distributed representations. In particular, we show that a mixture of products
requiring an exponentially larger number of parameters is needed to represent
the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table
A Nonparametric Ensemble Binary Classifier and its Statistical Properties
In this work, we propose an ensemble of classification trees (CT) and
artificial neural networks (ANN). Several statistical properties including
universal consistency and upper bound of an important parameter of the proposed
classifier are shown. Numerical evidence is also provided using various real
life data sets to assess the performance of the model. Our proposed
nonparametric ensemble classifier doesn't suffer from the `curse of
dimensionality' and can be used in a wide variety of feature selection cum
classification problems. Performance of the proposed model is quite better when
compared to many other state-of-the-art models used for similar situations
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