用格拉斯曼代数揭示神经网络的几何本质与推理机制
Neural Networks: According to the Principles of Grassmann Algebra
- 基于格拉斯曼代数构建神经网络的代数表示框架
- 通过幂等元与不变子空间实现关系路径的概率几何编码
- 为机器学习中的逻辑推理提供数学物理视角
本文探讨量子幂等元的代数结构及费米子的量子化，其生成的希尔伯特空间等于与李代数相关的格拉斯曼代数。由于幂等元承载所考虑代数的表示，它们在自然拓扑下形成代数簇和光滑流形。除了将数学物理与机器学习相联系的动机外，还表明利用幂等元及相应代数的不变子空间，这些表示可编码并可能以几何方式解释推理与关系路径的不确定性。
原文摘要 · Abstract (English)
In this paper, we explore the algebra of quantum idempotents and the quantization of fermions which gives rise to a Hilbert space equal to the Grassmann algebra associated with the Lie algebra. Since idempotents carry representations of the algebra under consideration, they form algebraic varieties and smooth manifolds in the natural topology. In addition to the motivation of linking up mathematical physics with machine learning, it is also shown that by using idempotents and invariant subspace of the corresponding algebras, these representations encode and perhaps provide a probabilistic interpretation of reasoning and relational paths in geometrical terms.
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