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Complex-Valued Boltzmann Manifold

Complex-Valued Boltzmann Manifold
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Author(s): Masaki Kobayashi (University of Yamanashi, Japan)
Copyright: 2009
Pages: 26
Source title: Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters
Source Author(s)/Editor(s): Tohru Nitta (National Institute of Advanced Industrial Science and Technology, Japan)
DOI: 10.4018/978-1-60566-214-5.ch001

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Abstract

Information geometry is one of the most effective tools to investigate stochastic learning models. In it, stochastic learning models are regarded as manifolds in the view of differential geometry. Amari applied it to Boltzmann Machines, which is one of the stochastic learning models. The purpose of this chapter is to apply information geometry to complex-valued Boltzmann Machines. First, we construct the complex-valued Boltzmann Machines. Next, the author describes information geometry. The author will know some important notions of information geometry, exponential families, mixture families, Kullback-Leibler divergence, connections, geodesics, Fisher metrics, potential functions and so on. Finally, they apply information geometry to complex-valued Boltzmann Machines. They will investigate the structure of complex-valued Boltzmann manifold and know the notions of the connections and Fisher metric. Moreover we will get an effective learning algorithm, what is called em algorithm, for complex-valued Boltzmann machines with hidden neurons.

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