用数论结构约束神经嵌入，让模型学习更可解释的3D表示。
Bhargava Cube--Inspired Quadratic Regularization for Structured Neural Embeddings
- 引入巴尔加瓦立方体的代数结构，对嵌入施加二次约束。
- 在MNIST上达99.46%准确率，嵌入自然按数字聚类且满足约束。
- 无需几何监督，通过可微代数先验提升可解释性，适合可解释AI研究者。
我们提出一种新颖的神经表示学习方法，引入数论中巴尔加瓦立方体的代数约束。传统深度学习在无结构潜在空间中学习表示，缺乏可解释性与数学一致性。本框架将输入数据映射到受约束的三维潜在空间，通过可微辅助损失函数引导嵌入满足由巴尔加瓦组合结构推导出的二次关系。该损失独立于分类目标，不依赖显式几何监督，兼容标准优化流程。在MNIST上实现99.46%准确率，生成的3D嵌入自然按数字类别聚类，并满足学习到的二次约束。这是首次将数论构造应用于神经表示学习，为在神经网络中引入结构化数学先验奠定基础。
原文摘要 · Abstract (English)
We present a novel approach to neural representation learning that incorporates algebraic constraints inspired by Bhargava cubes from number theory. Traditional deep learning methods learn representations in unstructured latent spaces lacking interpretability and mathematical consistency. Our framework maps input data to constrained 3-dimensional latent spaces where embeddings are regularized to satisfy learned quadratic relationships derived from Bhargava's combinatorial structures. The architecture employs a differentiable auxiliary loss function operating independently of classification objectives, guiding models toward mathematically structured representations. We evaluate on MNIST, achieving 99.46% accuracy while producing interpretable 3D embeddings that naturally cluster by digit class and satisfy learned quadratic constraints. Unlike existing manifold learning approaches requiring explicit geometric supervision, our method imposes weak algebraic priors through differentiable constraints, ensuring compatibility with standard optimization. This represents the first application of number-theoretic constructs to neural representation learning, establishing a foundation for incorporating structured mathematical priors in neural networks.
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