By Tohru Nitta
Contemporary study exhibits that complex-valued neural networks whose parameters (weights and threshold values) are all complicated numbers are actually important, containing features bringing approximately many major functions. Complex-Valued Neural Networks: using High-Dimensional Parameters covers the present state of the art theories and functions of neural networks with high-dimensional parameters resembling complex-valued neural networks, quantum neural networks, quaternary neural networks, and Clifford neural networks, that have been constructing lately. Graduate scholars and researchers will simply gather the elemental wisdom had to be on the vanguard of study, whereas practitioners will without difficulty take in the fabrics required for the functions.
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Extra resources for Complex-valued Neural Networks: Utilizing High-dimensional Parameters (Premier Reference Source)
IEICE Transactions on Fundamentals, E75-A(5), 531-536. Amari, S. (1994). Information geometry and manifolds of neural network. , Statistical Physics to Statistical Inference and Back, (pp. 113-138). Kluwer Academic Publishers. Amari, S. (1995). Information geometry of the EM and em algorithms for neural networks. Neural Networks, 8(9), 1379-1408. Amari, S. (1995). The EM algorithm and Information geometry in neural network learnings. Neural Computation, 7(1), 13-18. Amari, S. (1998). Natural gradient works efficiently in learning.
We denote p(x) = ∑ i ri (x) and q(x) = ∑ i′ri (x) . Then we get the equation (1 – t) p(x) + t q(x) = ∑ ((1 − t ) i + t i′)ri (x) . Therefore i i i the m-geodesic (1 – t) p(x) + t q(x) is contained in S again. We describe the m‑geodesic (t) the expectation parameter. (t ) = E[(1 − t ) p ( ) + tq ( )] = (1 − t ) E[ p( )] + t E[q( )] = (1 − t ) ( p) + t ( q). We can regard the m-geodesic as a straight line in the expectation coordinate system. Consider a submanifold M in the manifold S. Let the probability densities p(x) and q(x) be any probability densities belonging to M.
The point q is called the e-projection from the probability density p to the submanifold M. They are called Projection Theorem. Projection Theorem. Let S be the family of discrete distributions. Then the following theorems hold. 1. 2. Let M be an e-autoparallel submanifold in S. For any p ∈ S, the point q ∈ M such that minimizes KL(p,q) is given by the m-projection from the probability density p to the submanifold M (Figure 2 (1)). Let M be an m-autoparallel submanifold in S. For any p∈ S, the point q ∈ M such that minimizes KL(q, p) is given by the e-projection from the probability density p to the submanifold M (Figure 2 (2)).
Complex-valued Neural Networks: Utilizing High-dimensional Parameters (Premier Reference Source) by Tohru Nitta