🤖 AI Summary
This work addresses the challenge that existing graph embedding methods struggle to accurately recover the geodesic geometry of manifolds under non-uniform sampling or in the presence of spurious short-circuit edges. The authors propose a path ensemble based on maximum-entropy random walks, constructing a free-energy distance by aggregating over all k-step paths. In the short-time limit, this distance approximates the squared geodesic distance, effectively balancing local neighborhood structure with global geometry. By integrating Varadhan’s heat kernel formula with symmetric kernel Gram decomposition, the method establishes a precise connection between free-energy distances and kernel methods. Scalability is enhanced through landmark-based projection and diffusion-potential pseudo-time. Experiments demonstrate that the approach significantly outperforms current diffusion- or shortest-path-based techniques on both synthetic manifolds and single-cell datasets, exhibiting superior robustness in preserving geodesic structure under non-uniform sampling and branching trajectory scenarios.
📝 Abstract
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.