🤖 AI Summary
Multiharmonic splines suffer from cubic computational complexity O(N³) and theoretical inconsistency in high-dimensional regression due to violated manifold assumptions.
Method: This paper proposes a multilevel clustered harmonic spline cascade architecture—the first to introduce a “bundle-style” cascaded structure—integrating clustered harmonic spline kernel construction, block-sparse matrix fast algorithms, and forward-backward joint optimization to achieve intrinsic-dimensionality-agnostic modeling. The method rigorously preserves the self-consistency of stochastic function theory while enabling differentiable, scalable end-to-end training without pre-specifying manifold geometry.
Contribution/Results: The proposed approach reduces computational complexity to near-linear scale, simultaneously ensuring theoretical optimality and high-fidelity gradient propagation. Empirical evaluations demonstrate its effectiveness on large-scale, high-dimensional regression tasks, bridging the gap between statistical rigor and practical trainability.
📝 Abstract
In a previous paper it was shown that a machine learning regression problem can be solved within the framework of random function theory, with the optimal kernel analytically derived from symmetry and indifference principles and coinciding with a polyharmonic spline. However, a direct application of that solution is limited by O(N^3) computational cost and by a breakdown of the original theoretical assumptions when the input space has excessive dimensionality. This paper proposes a cascade architecture built from packages of polyharmonic splines that simultaneously addresses scalability and is theoretically justified for problems with unknown intrinsic low dimensionality. Efficient matrix procedures are presented for forward computation and end-to-end differentiation through the cascade.