A real trained GP-LVM's Riemannian and Finslerian latent geodesics converge on high-dimensional real datasets, but diverge in low-dimensional synthetic settings

Real Gaussian Process Latent Variable Models (GP-LVMs, with stochastic active-set inference) are fit to a real 256-dimensional font-contour dataset and a real 48-dimensional qPCR single-cell gene-expression dataset, comparing geodesic distances computed under the standard expected (Riemannian) pullback metric against a more general Finsler metric that keeps the full distribution of stochastic pullback metrics rather than averaging it [pouplin-etal-2023-finslerian-latent-distances] On both real high-dimensional datasets, the Riemannian/Finsler volume-measure ratio near the data stays small (about 10^-4) and the two metrics' geodesics agree closely, empirically confirming a proven O(1/D) convergence rate between the two metrics as ambient dimension D grows [pouplin-etal-2023-finslerian-latent-distances] The two metrics visibly diverge only in low-dimensional (3D) synthetic toy datasets (pinwheel, concentric circles on a sphere), and the same convergence pattern as the real datasets is also observed on real MNIST and FashionMNIST, showing the common practice of approximating the stochastic pullback metric by its expectation is a safe approximation in the real, high-dimensional regime even though it is not exact in general [pouplin-etal-2023-finslerian-latent-distances]