Osculating Contact Atlases: Low-Rank Jet-Bundle Layers for Scalable Higher-Order Geometric Neural Computation}

Abstract

The Osculating Jet Geometric Layer (OJGL) introduces a higher-order geometric alternative to conventional neural layers by modeling hidden representations with local contact data: value, Jacobian, and curvature. Each layer learns multiple latent contact poles and approximates the transformation around each pole with an order-2 jet. A low-rank-plus-diagonal Hessian factorization reduces quadratic complexity from O(KD out ​ D in 2 ​ ) to O(KD out ​ D in ​ R), while soft routing blends the local approximations into a global transformation. The resulting Osculating Jet Geometric Network (OJGN) uses residual blocks, normalization, curvature regularization, and optional C 1 overlap consistency to encourage neighboring jets to agree in value and derivative. Theoretical results establish smoothness, approximation bounds, Hessian representability, and stability. Pilot experiments achieve 96.62% on Digits, 97.78% on Wine, and 96.78% on Breast Cancer Wisconsin. OJGN improves over a same-width MLP on Digits, although parameter-matched MLPs remain competitive. The results support OJGN as a promising geometric neural primitive rather than a state-of-the-art accuracy claim. GitHub implementation: andrew-jeremy/Covariant-Osculating-Jet-Networks

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