Tutorials#

Each tutorial develops a geometric problem from its mathematical formulation through an executable GeoJAX solution and a visual report of the result.

Geometric deep learning#

Euclidean, spherical, and hyperbolic autoencoder latent spaces

Deep learning / Representation learning

Autoencoders with curved latent spaces

Train matched deterministic autoencoders on handwritten digits and compare Euclidean, spherical, and hyperbolic latents.
Karate Club graph embedded in Euclidean, spherical, and hyperbolic latent spaces

Graph learning / Node classification

Curvature-aware graph learning

Train matched intrinsic graph classifiers and compare Euclidean, spherical, and hyperbolic node representations.
Motor-imagery EEG covariance matrices and SPD latent representations

SPD learning / Real EEG

SPD prototype networks for motor-imagery EEG

Transform public sensorimotor EEG into covariance features and compare three geometric prototype heads.

Geometric computation#

Geodesics on a hyperboloid and in the Poincaré disk

Hyperbolic / Geometry

Geodesics on the hyperboloid

Trace unit-speed geodesics and compare the hyperboloid and Poincaré-disk views.
A wrapped geodesic viewed on a torus and in angular coordinates

Torus / Geometry

Wrapped geodesics on the torus

See why a smooth shortest path can jump across the edge of an angular chart.

Statistics and learning#

Four intrinsic and tangent-space classifiers compared on circular data

Sphere / Classification

Classification with manifold-valued predictors

Compare Fréchet centroids, geodesic k-NN, tangent logistic regression, and intrinsic-coordinate LDA.
Geodesic and local-linear regression curves with circular responses and bootstrap uncertainty

Sphere / Manifold response

Regression with circular responses

Fit parametric geodesic and local Fréchet regressions, then bootstrap a central circular response.
SPD covariance atoms, barycentric code matrix, and reconstruction errors

SPD / Representation learning

Barycentric covariance dictionaries

Replace linear combinations with intrinsic Karcher equations and learn positive-definite covariance atoms.
Robust circular locations, spatial depth, and mini-batch clustering

Sphere / Robust and scalable

Robust and scalable summaries

Contrast full, streaming, mini-batch, trimmed, and geodesic M-estimates on contaminated circular data.
Circular label propagation and graph-regularized response prediction

Sphere / Semi-supervised learning

Semi-supervised learning on the circle

Propagate sparse labels and smooth scalar responses over a geodesic-distance graph.
Three-dimensional point cloud, principal subspace, and optimization history

Grassmann / Data analysis

Principal component as a subspace

Recover a one-dimensional principal subspace from a synthetic point cloud on Gr(1, 3).
Hyperbolic observations in the Poincare disk, intrinsic clusters, and k-means objective

Hyperbolic / Clustering

Intrinsic k-means in hyperbolic space

Cluster points on the hyperboloid and inspect geodesic assignments through the Poincare disk.
Circular observations with intrinsic mean, median, enclosing ball, neighbors, and loss profiles

Sphere / Robust statistics

Robust location summaries on the circle

Compare the Fréchet mean, geometric median, enclosing ball, nearest neighbors, and a geodesic interpolation.
Seven clustering and quantization methods compared on a wrapped two-torus chart

Torus / Clustering

Comparing clustering methods

Contrast seven methods on periodic data whose clusters cross the boundaries of an angular chart.
Covariance ellipses, Fréchet means, and solver convergence

SPD / Statistics

Competing Fréchet means of SPD matrices

Compare log-Euclidean, affine-invariant, and Bures-Wasserstein covariance means.
Three-dimensional hand landmarks, intrinsic group means, and distance-based pose separation

Kendall shape / Real data

Shape-space inference for hand poses

Compute intrinsic means, classify SHREC'17 hand poses, and test the two pose populations with Fréchet ANOVA.
Two circular samples, their distance matrix, and permutation null distributions

Sphere / Inference

Two-sample inference on the circle

Compare Fréchet ANOVA, distance, energy, PSD-kernel MMD, and exact-Wasserstein permutation tests.
Covariance ellipses, principal geodesic coordinates, and a fitted response curve

SPD / Regression

PGA and regression with SPD predictors

Recover a covariance trajectory with PGA and fit a scalar response directly from manifold distances.
Spherical directions, periodic states, and a Product distance matrix

Product / Data adaptation

Adapting nested Product data

Convert mixed scientific representations into one validated pytree and reuse it across learning methods.
Two empirical measures on a circle and their exact optimal coupling

Sphere / Optimal transport

Exact transport on the circle

Solve a weighted transportation problem with geodesic costs and inspect feasibility and duality diagnostics.
A winding torus trajectory embedded by six dimension-reduction methods

Torus / Dimension reduction

Embedding a torus trajectory

Compare MDS, kernel PCA, Isomap, Sammon mapping, t-SNE, and PHATE on one closed winding.
Grassmann lines before and after projector-embedding metric learning

Grassmann / Supervised learning

Supervised metric learning on subspaces

Learn a task-specific metric on projector embeddings and compare it with intrinsic principal-angle distance.

Optimization and inverse problems#

Unit circle, Rayleigh objective, and solver convergence

Sphere / Optimization

Dominant eigenvector on the circle

Find the leading eigenvector of a symmetric matrix by optimizing the Rayleigh quotient on S1.
Initial and registered planar landmarks with solver convergence

SE(2) / Registration

Rigid landmark registration

Estimate a planar rotation and translation jointly by optimizing over the special Euclidean group.
Optimization paths, convergence histories, and iteration counts in a curved valley

Optimization / Solver choice

Comparing solvers in a curved valley

Contrast first-order, Hessian-based, and least-squares methods on one visual two-dimensional objective.