Riemann-Theta Boltzmann Machine
This work addresses the problem of enhancing neural network modeling capacity for researchers in machine learning, though it appears incremental as it builds on existing Boltzmann machine frameworks.
The authors introduced a general Boltzmann machine with continuous visible and discrete integer-valued hidden states, deriving an analytic probability density function using Riemann-Theta functions and enabling analytical computation of hidden state expectations for use as activation functions in feedforward networks.
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.