Optimisation Benchmarks

Wall-clock timing for Riemannian optimisation algorithms. cartan is compared against geoopt (RiemannianSGD) for gradient descent and against geomstats for the Frechet mean estimator. All times are median over 20 repetitions.

Riemannian Gradient Descent

RGD wall-clock time: cartan vs geoopt across dimensions on the sphere — Wall-clock time for 200 iterations of Riemannian gradient descent on the sphere, minimising the squared geodesic distance to a target point. cartan's RCG (dashed) shows the convergence advantage of conjugate gradient.

cartan provides three Riemannian optimisers:

Optimiser	cartan	geoopt equivalent
RGD (gradient descent)	`minimize_rgd`	`RiemannianSGD`
RCG (conjugate gradient)	`minimize_rcg`	n/a
RTR (trust region)	`minimize_rtr`	n/a

RCG and RTR are cartan-only; geoopt has no second-order Riemannian optimiser.

Optimisation Benchmarks

Riemannian Gradient Descent

Frechet Mean