这篇论文提出了一种流形学习新方法，用路径集成替代单条路径，在非均匀数据上比UMAP更忠实于测地几何。
EntroPath是一种新的流形学习方法，利用最大熵随机游走（MERW）聚合点之间k步路径全体来构建不相似度，避免局部随机游走的密度集中和最短路径的捷径噪声。其自由能不相似度在短时间极限下通过Varadhan热核公式收敛到平方测地距离。在合成流形和单细胞基准测试中，EntroPath一致优于扩散方法和最短路径方法，在非均匀采样密度流形和分支轨迹上优势明显，与UMAP、t-SNE本地结构指标持平。
EntroPath: Maximum Entropy Path Ensemble Embedding for Manifold Learning
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.