The manifold hypothesis, the idea that real-world high-dimensional data concentrates near a low-dimensional curved subspace, is foundational to modern machine learning. Many popular manifold learning methods such as UMAP, t-SNE, Isomap, and diff...

R-bloggers

An implementation of the Atlas-Learn algorithm from Robinett et al. (2025) in R, applied to the 2D sphere embedded in R³. The post explains the four-step algorithm: k-medoids partitioning, local PCA for tangent plane estimation, quadratic regression for curvature coefficients, and ellipsoidal domain construction. Full R code is provided for building the atlas data structure, computing geodesics via quasi-Euclidean retraction, and validating the learned manifold through great circle tracing, radius of curvature recovery, and the Gauss-Bonnet Theorem. The code was developed with Claude Sonnet 4.3 inside RStudio's AI Assistant.

The Atlas-Learn Approach to the Manifold Hypothesis