Hyperbolic Gradient Operator and Hyperbolic Back-Propagation Learning Algorithms
Hyperbolic Gradient Operator and Hyperbolic Back-Propagation Learning Algorithms
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DOI：
10.1109/tnnls.2017.2677446
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发表时间：
2018-05
影响因子：
10.4
通讯作者：
T. Nitta;Y. Kuroe
中科院分区：
文献类型：
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作者：
T. Nitta;Y. Kuroe
本文首先将复变函数中定义的Wirtinger导数推广到双曲函数，并利用它导出了产生最速下降方向的双曲梯度算子，然后利用双曲梯度算子导出了多层双曲神经网络的双曲反向传播学习算法.结果表明，使用Wirtinger导数减少了一半的学习算法的推导所需的努力，简化了学习算法的表示，并使其计算机程序更容易编码。此外，我们讨论了派生的双曲线BP规则和复值反向传播学习规则（Complex-BP）之间的差异。最后，我们对所提出的学习算法进行了实验。结果发现，即使使用完全激活函数，双曲BP学习算法的收敛速度也很高，并发现具有分裂型双曲激活函数的双曲神经网络的双曲BP学习算法具有学习双曲旋转的能力。
In this paper, we first extend the Wirtinger derivative which is defined for complex functions to hyperbolic functions, and derive the hyperbolic gradient operator yielding the steepest descent direction by using it. Next, we derive the hyperbolic backpropagation learning algorithms for some multilayered hyperbolic neural networks (NNs) using the hyperbolic gradient operator. It is shown that the use of the Wirtinger derivative reduces the effort necessary for the derivation of the learning algorithms by half, simplifies the representation of the learning algorithms, and makes their computer programs easier to code. In addition, we discuss the differences between the derived Hyperbolic-BP rules and the complex-valued backpropagation learning rule (Complex-BP). Finally, we make some experiments with the derived learning algorithms. As a result, we find that the convergence rates of the Hyperbolic-BP learning algorithms are high even if the fully activation functions are used, and discover that the Hyperbolic-BP learning algorithm for the hyperbolic NN with the split-type hyperbolic activation function has an ability to learn hyperbolic rotation as its inherent property.