EntroPath Improves Manifold Learning with Maximum Entropy Paths

Manifold learning techniques, which aim to uncover the underlying geometric structure of high-dimensional data, often rely on graph-based embeddings. However, current methods face limitations: those based on locally normalized random walks can over-concentrate in dense regions, while shortest-path distances are vulnerable to misleading "shortcut" edges in the data graph. These issues can lead to an inaccurate representation of the true geodesic distances within the data. Researchers have introduced EntroPath, a novel manifold learning method designed to address these challenges. EntroPath constructs its dissimilarities using maximum entropy random walks (MERW), which consider the full ensemble of k-step paths between data points, rather than relying on single trajectories or local neighborhoods. This path-ensemble approach allows for a more robust and faithful recovery of the data's geodesic geometry. EntroPath's free-energy dissimilarity is shown to converge to squared geodesic distance in the short-time limit. The method demonstrates superior performance over diffusion- and shortest-path-based techniques, particularly on manifolds with non-uniform sampling densities and complex branching structures. It also remains competitive with popular neighborhood-preserving embeddings like UMAP and t-SNE on local structure metrics, offering scalable extensions for larger datasets.