Hypercomplex algebras have recently been gaining prominence in the field of deep learning owing to the advantages of their division algebras over real vector spaces and their superior results when dealing with multidimensional signals in real-world 3D and 4D paradigms. This article provides a foundational framework that serves as a road map for understanding why hypercomplex deep learning methods are so successful and how their potential can be exploited. Such a theoretical framework is described in terms of inductive bias, i.e., a collection of assumptions, properties, and constraints that are built into training algorithms to guide their learning process toward more efficient and accurate solutions. We show that it is possible to derive specific inductive biases in the hypercomplex domains, which extend complex numbers to encompass diverse numbers and data structures. These biases prove effective in managing the distinctive properties of these domains as well as the complex structures of multidimensional and multimodal signals. This novel perspective for hypercomplex deep learning promises to both demystify this class of methods and clarify their potential, under a unifying framework, and in this way, promotes hypercomplex models as viable alternatives to traditional real-valued deep learning for multidimensional signal processing.

中文翻译：

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揭秘超复杂：超复杂深度学习中的归纳偏差[超复杂信号和图像处理]


超复代数最近在深度学习领域越来越受到关注，因为它们的除法代数相对于实向量空间的优势以及在处理现实世界 3D 和 4D 范式中的多维信号时的优越结果。本文提供了一个基础框架，可作为理解超复杂深度学习方法为何如此成功以及如何挖掘其潜力的路线图。这样的理论框架是用归纳偏差来描述的，即内置于训练算法中的假设、属性和约束的集合，以指导其学习过程获得更有效和更准确的解决方案。我们证明，可以在超复数域中导出特定的归纳偏差，从而将复数扩展到包含不同的数字和数据结构。事实证明，这些偏差可以有效管理这些领域的独特属性以及多维和多模态信号的复杂结构。这种超复杂深度学习的新颖视角有望在统一框架下揭开此类方法的神秘面纱并阐明其潜力，从而促进超复杂模型成为多维信号处理的传统实值深度学习的可行替代方案。