12,543 research outputs found
Continuous Restricted Boltzmann Machines
Restricted Boltzmann machines are a generative neural network. They summarize their input data to build a probabilistic model that can then be used to reconstruct missing data or to classify new data. Unlike discrete Boltzmann machines, where the data are mapped to the space of integers or bitstrings, continuous Boltzmann machines directly use floating point numbers and therefore represent the data with higher fidelity. The primary limitation in using Boltzmann machines for big-data problems is the efficiency of the training algorithm. This paper describes an efficient deterministic algorithm for training continuous machines
Neural Networks retrieving Boolean patterns in a sea of Gaussian ones
Restricted Boltzmann Machines are key tools in Machine Learning and are
described by the energy function of bipartite spin-glasses. From a statistical
mechanical perspective, they share the same Gibbs measure of Hopfield networks
for associative memory. In this equivalence, weights in the former play as
patterns in the latter. As Boltzmann machines usually require real weights to
be trained with gradient descent like methods, while Hopfield networks
typically store binary patterns to be able to retrieve, the investigation of a
mixed Hebbian network, equipped with both real (e.g., Gaussian) and discrete
(e.g., Boolean) patterns naturally arises. We prove that, in the challenging
regime of a high storage of real patterns, where retrieval is forbidden, an
extra load of Boolean patterns can still be retrieved, as long as the ratio
among the overall load and the network size does not exceed a critical
threshold, that turns out to be the same of the standard
Amit-Gutfreund-Sompolinsky theory. Assuming replica symmetry, we study the case
of a low load of Boolean patterns combining the stochastic stability and
Hamilton-Jacobi interpolating techniques. The result can be extended to the
high load by a non rigorous but standard replica computation argument.Comment: 16 pages, 1 figur
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
Event-Driven Contrastive Divergence for Spiking Neuromorphic Systems
Restricted Boltzmann Machines (RBMs) and Deep Belief Networks have been
demonstrated to perform efficiently in a variety of applications, such as
dimensionality reduction, feature learning, and classification. Their
implementation on neuromorphic hardware platforms emulating large-scale
networks of spiking neurons can have significant advantages from the
perspectives of scalability, power dissipation and real-time interfacing with
the environment. However the traditional RBM architecture and the commonly used
training algorithm known as Contrastive Divergence (CD) are based on discrete
updates and exact arithmetics which do not directly map onto a dynamical neural
substrate. Here, we present an event-driven variation of CD to train a RBM
constructed with Integrate & Fire (I&F) neurons, that is constrained by the
limitations of existing and near future neuromorphic hardware platforms. Our
strategy is based on neural sampling, which allows us to synthesize a spiking
neural network that samples from a target Boltzmann distribution. The recurrent
activity of the network replaces the discrete steps of the CD algorithm, while
Spike Time Dependent Plasticity (STDP) carries out the weight updates in an
online, asynchronous fashion. We demonstrate our approach by training an RBM
composed of leaky I&F neurons with STDP synapses to learn a generative model of
the MNIST hand-written digit dataset, and by testing it in recognition,
generation and cue integration tasks. Our results contribute to a machine
learning-driven approach for synthesizing networks of spiking neurons capable
of carrying out practical, high-level functionality.Comment: (Under review
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
Discriminative conditional restricted Boltzmann machine for discrete choice and latent variable modelling
Conventional methods of estimating latent behaviour generally use attitudinal
questions which are subjective and these survey questions may not always be
available. We hypothesize that an alternative approach can be used for latent
variable estimation through an undirected graphical models. For instance,
non-parametric artificial neural networks. In this study, we explore the use of
generative non-parametric modelling methods to estimate latent variables from
prior choice distribution without the conventional use of measurement
indicators. A restricted Boltzmann machine is used to represent latent
behaviour factors by analyzing the relationship information between the
observed choices and explanatory variables. The algorithm is adapted for latent
behaviour analysis in discrete choice scenario and we use a graphical approach
to evaluate and understand the semantic meaning from estimated parameter vector
values. We illustrate our methodology on a financial instrument choice dataset
and perform statistical analysis on parameter sensitivity and stability. Our
findings show that through non-parametric statistical tests, we can extract
useful latent information on the behaviour of latent constructs through machine
learning methods and present strong and significant influence on the choice
process. Furthermore, our modelling framework shows robustness in input
variability through sampling and validation
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