On Clifford neurons and Clifford multi-layer perceptrons

Area of Science:

• * Computational neuroscience
• * Geometric deep learning
• * Clifford algebra

Background:

• * Standard neural networks struggle with processing complex geometric entities.
• * Clifford algebra offers a powerful framework for representing and manipulating geometric data.
• * Existing Clifford neural network models lack a unified theoretical basis and training methods.

Purpose of the Study:

• * To establish a theoretical foundation for Clifford neural computation.
• * To introduce novel neural architectures, such as Spinor Clifford neurons, for geometric processing.
• * To develop a unified training rule and explore activation functions for Clifford Multilayer Perceptrons (MLPs).

Main Methods:

• * Application of Clifford algebra framework to design neural architectures.
• * Introduction of isomorphic neurons and isomorphic representations.
• * Development of a unified training rule for Clifford MLPs.
• * Detailed analysis of activation functions for 2D Clifford algebras.

Main Results:

• * Demonstrated benefits of Clifford-based computation over standard real-valued networks.
• * Introduced a new class of Spinor Clifford neurons.
• * Provided a sound theoretical basis for Clifford neural computation.
• * Presented a unified training rule applicable to Clifford MLPs.
• * Explored activation functions for all two-dimensional Clifford algebras.

Conclusions:

• * Clifford algebra provides a robust framework for advanced neural network design.
• * The proposed Spinor Clifford neurons and unified training rule advance Clifford neural computation.
• * This work lays the groundwork for future research in geometric deep learning and Clifford neural networks.