System and method for adaptive geometric diffusion projection onto manifolds

Abstract

A system and method for adaptive geometric diffusion projection enables mapping of heterogeneous high-dimensional representations onto a shared low-dimensional manifold without neural network training. The system maintains landmark points in source spaces and computes their spectral coordinates through graph Laplacian eigen decomposition based on semantic similarities. New input points are projected via harmonic extension, computing weighted interpolations of nearby landmark spectral coordinates. A geometric optimization process refines positions while maintaining manifold constraints through tangent space projections. The system continuously monitors geometric invariants including principal angles, spectral gaps, and curvature distributions. When invariants exceed thresholds, targeted adaptations occur: spectral basis updates using warm-started iterations, landmark set augmentation in high-residual regions, or parameter adjustments. The approach supports logarithmic computational scaling, enables streaming operation on continuous data, and handles multimodal inputs through reliability-weighted consensus. The system maintains projection quality indefinitely through continuous geometric monitoring and local adaptations.