Yang-Mills-Higgs: 非可缩空间上二元标签的几何理论
Yang-Mills-Higgs: A Geometric Theory of Binary Labels on Non-Contractible Spaces
- Ness Digital Engineering
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
将流形上的二元分类重新表述为Yang-Mills-Higgs变分问题，通过拓扑障碍和Bogomolny不等式实现分类器与规范背景的联合优化，并揭示了曲率与Transformer注意力的对应关系。
AI中文摘要：
我们将流形M上的二元分类重新表述为一个Yang-Mills-Higgs变分问题。标记数据被编码为一个从M的基本群胚到单对象群胚B(Z_2)的函子，其在H^1(M, Z_2)中的单值类是实现分类器为符号函数的拓扑障碍。分类器截面和联络在硬数据条件下共同最小化Yang-Mills-Higgs能量：物质部分承载分类内容，而Yang-Mills部分在每个拓扑类中由Bogomolny不等式给出下界，并选择规范背景。这恢复了伴随论文中作为可缩底空间、平坦联络约化的调和插值。两个结构性结果随之而来。首先，所选联络的曲率2-形式与Transformer注意力有精确的对应：它是注意力双线性形式的反对称部分，曲率的阿贝尔/非阿贝尔分裂对应于单头/多头注意力分裂。其次，环面上的XOR问题由双Möbius丛的协变调和截面以闭式解出，最小能量2*pi^2经数值验证达到机器精度，而训练在相同数据上的MLP找到了一个忽略环面识别的结构不同的边界。实例演算了一个示例阶梯（圆、环面、S^2 Dirac单极子、S^4 BPST瞬子）；在两点情形下证明了物质部分的邻近缩放定理。
英文摘要：
We reformulate binary classification on a manifold M as a Yang-Mills-Higgs variational problem. Labelled data is encoded as a functor from the fundamental groupoid of M to the one-object groupoid B(Z_2), whose monodromy class in H^1(M, Z_2) is a topological obstruction to realising the classifier by a sign function. The classifier-section and the connection jointly minimise a Yang-Mills-Higgs energy subject to hard data conditions: the matter sector carries the classification content, while the Yang-Mills sector is bounded below in each topological class by the Bogomolny inequality and selects the gauge background. This recovers the companion paper's harmonic interpolation as the contractible-base, flat-connection reduction. Two structural payoffs follow. First, the curvature 2-form of the selected connection supplies a dictionary with transformer attention: by the plaquette formula it is a pairwise, antisymmetric, algebra-valued form on tangent directions, which we match - as a stipulated dictionary, not a derived identity - to the antisymmetric component of the attention bilinear; the abelian/non-abelian split of curvature corresponds to the single-head/multi-head split of attention. Second, XOR on the torus is realised by the covariantly harmonic section of the double-Mobius bundle, which the variational selector picks out: solving the regularised capacitance system on all four flat Z_2-bundles gives a strict energy ordering favouring the double-Mobius class, whereas an MLP trained on the same data finds a structurally different boundary that ignores the toroidal identifications. Worked examples run an example ladder (circle, torus, S^2 monopole, S^4 instanton). The connection is throughout either flat or pinned to the Bogomolny moduli, so curvature is data-decoupled by construction; whether freeing it makes curvature respond to label proximity is posed as an open question.
