Fractional-order Jacobian Matrix Differentiation and Its Application in Artificial Neural Networks
This work addresses a theoretical gap for researchers in optimization and deep learning, enabling practical use of fractional-order differentiation in neural networks, though it appears incremental as it builds on existing integer-order methods.
The paper tackled the lack of a fractional-order matrix differentiation method compatible with automatic differentiation by proposing fractional-order Jacobian matrix differentiation (J^α), enabling fractional-order chain rules and Autograd technology, and demonstrated superior performance in deep learning experiments with improved test set metrics and efficiency.
Fractional-order differentiation has many characteristics different from integer-order differentiation. These characteristics can be applied to the optimization algorithms of artificial neural networks to obtain better results. However, due to insufficient theoretical research, at present, there is no fractional-order matrix differentiation method that is perfectly compatible with automatic differentiation (Autograd) technology. Therefore, we propose a fractional-order matrix differentiation calculation method. This method is introduced by the definition of the integer-order Jacobian matrix. We denote it as fractional-order Jacobian matrix differentiation (${\bf{J}^α}$). Through ${\bf{J}^α}$, we can carry out the matrix-based fractional-order chain rule. Based on the Linear module and the fractional-order differentiation, we design the fractional-order Autograd technology to enable the use of fractional-order differentiation in hidden layers, thereby enhancing the practicality of fractional-order differentiation in deep learning. In the experiment, according to the PyTorch framework, we design fractional-order Linear (FLinear) and replace nn.Linear in the multilayer perceptron with FLinear. Through the qualitative analysis of the training set and validation set $Loss$, the quantitative analysis of the test set indicators, and the analysis of time consumption and GPU memory usage during model training, we verify the superior performance of ${\bf{J}^α}$ and prove that it is an excellent fractional-order gradient descent method in the field of deep learning.