Catalin Vasii | Searcharxiv

arXiv2026

From Gradient Descent to Potential Theory: A Geometric Dictionary for Binary Classification

We propose a dictionary between binary classification in machine learning and classical potential theory on vector bundles. Classifiers are parallel sections of vector bundles over the data space; training labels become Dirichlet boundary conditions; the kernel of an RKHS interpolant is the Green's function of an elliptic operator; and backpropagation is the flat-geometry limit of an exact geometric problem. The central identification: for L = (Delta + kappa^2)^nu with nu > n/2, the L-harmonic interpolation problem - find the minimum-H^nu-norm classifier satisfying Lf = 0 away from the data with prescribed values at the training points - is precisely what RKHS interpolation and kernel ridge regression already solve. Kimeldorf-Wahba (1971) and Lindgren-Rue-Lindstrom (2011) supply the analytic content; what we add is the potential-theoretic reading: the kernel is the Green's function of L, the coefficients are electrostatic capacitances, and the decision boundary is the zero equipotential. The condition nu > n/2 excludes the Dirichlet energy (nu = 1) in dimension n >= 2, where point data has zero capacity. The hard-margin SVM is a margin-constrained variant. For finite data on any smooth manifold, flat O(2) solutions always exist. Over non-contractible bases the geometry becomes load-bearing: the parity invariant on S^1 and the Euler class on S^2 are obstructions no classifier can evade. Code is available on GitHub.

math.DG↗

arXiv2026

Yang-Mills-Higgs: A Geometric Theory of Binary Labels on Non-Contractible Spaces

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.

math.DG↗