Linear-cost Polyharmonic Spline Interpolation of Arbitrary Degree

What the math gives to ML

The paper provides a scalable way to apply conditionally positive-definite polyharmonic radial kernels and solve their constrained interpolation systems without forming a dense matrix. The transferable asset is a combination of an FMM matrix-vector product, polynomial null-space constraints, and a sparse Vecchia/Matérn precision preconditioner that makes projected conjugate gradients converge in roughly a constant number of iterations. This can become a million-anchor neural-field or point-cloud decoder whose geometric kernel component is exact up to FMM tolerance rather than a dense learned RBF layer. The most direct first experiment is a hybrid coordinate network in which a small MLP supplies residual detail while a PHS head handles global interpolation from many training anchors.