Harnessing Data Asymmetry: Manifold Learning in the Finsler World

Many datasets carry an intrinsic directionality: citations point backward in time, cells differentiate along lineages, and traffic follows preferred routes. Spectral embedding methods, including most of their extensions to directed graphs, discard this information: they symmetrize the data and map it into a Euclidean space where asymmetry cannot be represented. We instead model directed data as sampled from a Finsler manifold, whose distance depends on the direction of travel, and study the kernel operator built from this asymmetric distance. Through a moment expansion of this operator, we show that its symmetric and antisymmetric parts separate geometry from direction. As the bandwidth of the kernel vanishes, the symmetric part converges to a weighted Laplacian, recovering diffusion maps in the Riemannian case, while the antisymmetric part converges to a first-order transport operator that encodes the directionality. We prove that the corresponding graph operators, built from finitely many samples, converge uniformly and almost surely to these limits. For Randers metrics, this vector field is explicit and yields an embedding algorithm recovering both the manifold structure, from the spectrum of the symmetric part, and the underlying drift. We illustrate the approach on synthetic directed graphs and point-clouds.

We introduce EntroPath, a manifold learning method that recovers geodesic geometry from data graphs through ensembles of diffusion paths. Many existing graph-based embeddings rely either on locally normalised random walks or on shortest-path distances. The former can concentrate diffusion in densely sampled regions, while the latter are sensitive to spurious shortcut edges in the graph. EntroPath instead builds its dissimilarities from the maximum entropy random walk (MERW), which aggregates the full ensemble of k-step paths between points rather than relying on any single trajectory. We show that the resulting free-energy dissimilarity converges to squared geodesic distance in the short-time limit, via Varadhan's heat-kernel formula. The diffusion depth k interpolates smoothly between local neighbourhood structure and global manifold geometry, and the symmetrised kernel admits an exact Gram factorisation connecting EntroPath to kernel methods. We further provide scalable extensions via landmark projection and diffusion-potential pseudotime. Across synthetic manifolds and single-cell benchmarks, EntroPath consistently matches or outperforms diffusion- and shortest-path-based methods, while remaining competitive with neighbourhood-preserving embeddings (UMAP, t-SNE) on local-structure metrics. Its gains are most pronounced on manifolds with non-uniform sampling density and well-separated branching trajectories, where path-ensemble diffusion more faithfully preserves the underlying geodesic geometry.

The Gini Multidimensional Scaling (Gini MDS) framework extends the Euclidean multidimensional scaling. We introduce a Gini pseudo-distance based on values and their ranks that depends on a fine-tunable hyperparameter. This pseudo-distance allows flexible exploration of latent configurations, enabling embeddings that best match observed dissimilarities. The Gini MDS is shown to be robust to noise and outliers, making it well-suited for real-world applications. We provide experiments on 16 UCI datasets with outliers and on MNIST images with noise to show that the Gini MDS outperforms the Euclidean MDS on noisy data. Finally, a tensor-based implementation in PyTorch provides GPU acceleration and efficient computation compared to the standard MDS of the sklearn library.