New Method Offers Scalable Geometric Modeling of High-Dimensional Data.

High-dimensional datasets often exhibit intrinsic low-dimensional structures, but traditional methods for estimating their geometry, such as those based on graphs and kernels, struggle with scalability as data size and dimension increase. This research introduces Riemannian metric matching, a novel denoising probabilistic framework. The framework utilizes neural networks to learn the Riemannian geometry of data by specifically learning the carr\'e du champ operator. This operator, derived from diffusion geometry, provides access to a comprehensive toolkit for various machine learning and statistical tasks. A key insight is that the carr\'e du champ operator can be expressed as a conditional expectation over random data perturbations, allowing for sample-wise training and constant-cost, amortized inference without explicit kernel construction. Empirically, metric matching achieves comparable or superior accuracy to k-NN-based diffusion geometry estimators while being significantly faster (up to 400x) and supporting graph-free geometric analysis on challenging high-dimensional data like images.