Several models of feed-forward complex-valued neural networks have been proposed, and those with split and polar-represented activation functions have been mainly studied. Neural networks with split activation functions are relatively easy to analyze, but complex-valued neural networks with polar-represented functions have many applications but are difficult to analyze. In previous research, Nitta proved the uniqueness theorem of complex-valued neural networks with split activation functions. Subsequently, he studied their critical points, which caused plateaus and local minima in their learning processes. Thus, the uniqueness theorem is closely related to the learning process. In the present work, we first define three types of reducibility for feed-forward complex-valued neural networks with polar-represented activation functions and prove that we can easily transform reducible complex-valued neural networks into irreducible ones. We then prove the uniqueness theorem of complex-valued neural networks with polar-represented activation functions.

Complex-valued Hopfield associative memory (CHAM) is one of the most promising neural network models to deal with multilevel information. CHAM has an inherent property of rotational invariance. Rotational invariance is a factor that reduces a network's robustness to noise, which is a critical problem. Here, we proposed complex-valued bipartite auto-associative memory (CBAAM) to solve this reduction in noise robustness. CBAAM consists of two layers, a visible complex-valued layer and an invisible real-valued layer. The invisible real-valued layer prevents rotational invariance and the resulting reduction in noise robustness. In addition, CBAAM has high parallelism, unlike CHAM. By computer simulations, we show that CBAAM is superior to CHAM in noise robustness. The noise robustness of CHAM decreased as the resolution factor increased. On the other hand, CBAAM provided high noise robustness independent of the resolution factor.

This paper presents a Complex-Valued Neural Network-based sound localization method. The proposed approach uses two microphones to localize sound sources in the whole horizontal plane. The method uses time delay and amplitude difference to generate a set of features which are then classified by a Complex-Valued Multi-Layer Perceptron. The advantage of using complex values is that the amplitude information can naturally masks the phase information. The proposed method is analyzed experimentally with regard to the spectral characteristics of the target sounds and its tolerance to noise. The obtained results emphasize and confirm the advantages of using Complex-Valued Neural Networks for the sound localization problem in comparison to the traditional Real-Valued Neural Network model.

Complex-valued Associative Memory (CAM) is an advanced model of Hopfield Associative Memory. The CAM is based on multi-state neurons and has the high ability of representation. Lee proposed gradient descent learning for the CAM to improve the storage capacity. It is based on only the phases of input signals. In this paper, we propose another type of gradient descent learning based on both the phases and the amplitude. The proposed learning method improves the noise robustness and accelerates the learning speed.