K-Splanifolds: Advancing General Purpose Regression with Linear-Time Parametric Spline Manifolds

Many regression tasks in geometric modeling, deformation control, and reduced simulation require fast, stable mappings from controls in [0, 1]K to outputs in RN . K-Splanifolds are structured parametric regression manifolds for this regime and can act as a practical alternative to small coordinate MLPs. They construct the map with a cubic spline spine plus transverse deviation components that separate global progression from local variation. The approach has O(KN ) storage and evaluation, remains differentiable, supports direct linear least-squares fitting, and includes smooth tangent regularization with bounded extrapolation. In reported baselines (N =256), runtime at K=256 is 21.74 μs/query versus 779.06 μs for RBF (C=256), while remaining in the same latency regime as a small coordinate MLP (8.29 μs). At comparable RMSE, entropy-coded bits per parameter are substantially lower than MLPs (smooth cubic: 3.71 vs. 5.94; noisy irregular: 2.97 vs. 4.70), indicating higher information density at similar loss. These properties may also be useful in selected NLP and LLM components, but that transfer requires dedicated empirical validation.