Wirtinger Calculus and Complex Natural Gradient Algorithm

This paper presents an extension of the real tangent spaces on the form of the Wirtinger (complex) calculus in order to find a generalized expression for the natural complex gradient. It is presented an expression for the natural gradient, which is valid to a general coordinate system, expressed in a complex basis, allowing the use of of the fully complex notation. The equivalence of complex and real representations is done through the complex structure that real spaces may have, allowing the change of the used basis. Some examples typical applications are presented, illustrating the potential of the proposal.