Metacognitive Octonion-Valued Neural Networks as They Relate to Time Series Analysis

New results on bifurcation for fractional-order octonion-valued neural networks involving delays

This work chiefly explores fractional-order octonion-valued neural networks involving delays. We decompose the considered fractional-order delayed octonion-valued neural networks into equivalent real-valued systems via Cayley-Dickson construction. By virtue of Lipschitz condition, we prove that the solution of the considered fractional-order delayed octonion-valued neural networks exists and is unique. By constructing a fairish function, we confirm that the solution of the involved fractional-order delayed octonion-valued neural networks is bounded. Applying the stability theory and basic bifurcation knowledge of fractional order differential equations, we set up a sufficient condition remaining the stability behaviour and the appearance of Hopf bifurcation for the addressed fractional-order delayed octonion-valued neural networks. To illustrate the justifiability of the derived theoretical results clearly, we give the related simulation results to support these facts. Simultaneously, the bifurcation plots are also displayed. The established theoretical results in this work have important guiding significance in devising and improving neural networks.

Multivariate Time Series Prediction Based on Temporal Change Information Learning Method

In the multivariate time series prediction tasks, the impact information of all nonpredictive time series on the predictive target series is difficult to be extracted at different time stages. Through the emphasis on optimal-related sequences in the target series, the deep learning model with the attention mechanism achieves a good predictive performance. However, temporal change information in the objective function and optimization algorithm is completely ignored in these models. To this end, a temporal change information learning (CIL) method is proposed in this article. First, mean absolute error (MAE) and mean squared error (MSE) losses are contained in the objective function to evaluate different amplitude errors. Meanwhile, the second-order difference technology is used in the correlation terms of the objective function to adaptively capture the impact of the abrupt and slow change information in each series on the target series. Second, the long short-term memory (LSTM) network with the transformation mechanism is used in the method so that temporal dependence information can be fully extracted (i.e., avoiding the supersaturation region). Third, to effectively obtain the optimal model parameters, the current and historical moment estimation information is adaptively memorized without the introduction of additional hyperparameters, and therefore, the acquisition ability of temporal change information in the error gradient flow is greatly enhanced by the proposed optimization algorithm. Finally, three datasets with different scales are used to verify the advantages of the CIL method in computational overhead and prediction effect.

The dynamics of octonion-valued neutral type high-order Hopfield neural networks with D operator

In this paper, the existence, uniqueness and global exponential stability of pseudo almost periodic solutions for a class of octonion-valued neutral type high-order Hopfield neural network models with D operator are established by using the Banach fixed point theorem and differential inequality techniques. Compared with most existing models, in this class of networks, all connection weights and activation functions are assumed to be octonion-valued functions except for time delays. And unlike most of the existing methods of studying octonion-valued neural networks, our method is a non-decomposition method, that is, the method of directly studying octonion-valued systems. The results and methods in this paper are new. In addition, an example and its numerical simulation are given to illustrate the feasibility of our results.

Fast Algorithms for Deep Octonion Networks

This brief presents the results of a study of the possibilities of reducing the arithmetic complexity of computing basic operations in octonionic neural networks and also proposes new algorithmic solutions for efficiently performing these operations. Here, we primarily mean the operation of multiplying octonions, the operation of computing the dot product of two octonion-valued vectors, and the operation of multiple multiplications of an octonion by several other octonions. In order to reduce the computational complexity of these operations, it is proposed to use the fast Walsh-Hadamard transform, which is well known in digital signal processing. Using this transform reduces the number of multiplications and additions of real numbers required to perform computations. Thus, the use of the proposed algorithms will speed up computations in octonion-valued neural networks.