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Modelling conditional probabilities with Riemann-Theta Boltzmann Machines
The probability density function for the visible sector of a Riemann-Theta
Boltzmann machine can be taken conditional on a subset of the visible units. We
derive that the corresponding conditional density function is given by a
reparameterization of the Riemann-Theta Boltzmann machine modelling the
original probability density function. Therefore the conditional densities can
be directly inferred from the Riemann-Theta Boltzmann machine.Comment: 7 pages, 3 figures, in proceedings of the 19th International Workshop
on Advanced Computing and Analysis Techniques in Physics Research (ACAT 2019
Riemann-Theta Boltzmann Machine
A general Boltzmann machine with continuous visible and discrete integer
valued hidden states is introduced. Under mild assumptions about the connection
matrices, the probability density function of the visible units can be solved
for analytically, yielding a novel parametric density function involving a
ratio of Riemann-Theta functions. The conditional expectation of a hidden state
for given visible states can also be calculated analytically, yielding a
derivative of the logarithmic Riemann-Theta function. The conditional
expectation can be used as activation function in a feedforward neural network,
thereby increasing the modelling capacity of the network. Both the Boltzmann
machine and the derived feedforward neural network can be successfully trained
via standard gradient- and non-gradient-based optimization techniques.Comment: 29 pages, 11 figures, final version published in Neurocomputin
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