[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84122-en":3,"doc-seo-84122-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84122,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","EntroPath: Maximum Entropy Path Ensemble Embedding for Manifold Learning","EntroPath presents a manifold learning approach that reconstructs geodesic geometry from data graphs using ensembles of diffusion paths. Unlike embeddings based on locally normalized random walks or fragile shortest-path distances, EntroPath defines dissimilarities via maximum entropy random walks (MERW) that aggregate all k-step paths rather than a single trajectory. The free-energy dissimilarity converges to squared geodesic distance in the short-time limit using Varadhan’s heat-kernel formula, enabling smooth interpolation between local and global geometry and scalable landmark and pseudotime extensions.","arXiv :2607 .06497v 1 [ cs .LG] 7 Jul 2026  \nENTROPATH: MAXIMUM ENTROPY PATH ENSEMBLE EMBEDDING FOR MANIFOLD LEARNING  \nA PREPRINT  \nPrzemysław Rola  \nDepartment of Mathematics, Krakow University of Economics  \n[przemyslaw.rola@outlook.com](przemyslaw.rola@outlook.com)  \nJuly 8, 2026  \nABSTRACT  \nWe 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.  \nEntroPath 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.  \nWe 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 diffusionand 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.  \nKeywords Manifold Learning · Dimensionality Reduction · Geodesic Distance · Maximum Entropy Random Walk · Diffusion Geometry · Path Ensemble · Single-Cell Trajectory Inference  \n1 Introduction  \nMotivation. Modern high-dimensional datasets—single-cell transcriptomics, molecular dynamics trajectories, and image manifolds—typically concentrate near a low-dimensional Riemannian manifold embedded in ambient space. The intrinsic geometry of such a manifold is characterised by its geodesic distances, and preserving this geometry in a low-dimensional representation is a central goal of manifold learning.  \nExisting methods approach this problem in different ways. Graph shortest-path methods, exemplified by Isomap [27], estimate geodesic distances as shortest paths through a neighbourhood graph. This single-path estimate is fragile: a single short-circuit edge between geodesically distant regions can corrupt the resulting distances and distort the global embedding [1] . Moreover, theoretical recovery guarantees require restrictive assumptions, such as convex parameter domains, that are violated by branching or holed manifolds, producing strong distortions around the undersampled regions [5, 9] . Other approaches, including t-SNE [29] and UMAP [17], focus primarily on preserving local neighbourhood relationships through probabilistic objectives rather than explicitly approximating manifold geodesics. While often effective for visualisation, these methods do not directly encode global geodesic structure. Diffusion and spectral methods instead aggregate information over an ensemble of graph paths through a diffusion operator. By integrating many paths rather than relying on a single shortest path, they are less sensitive to individual graph errors and provide a natural connection to the manifold’s intrinsic geometry. Within this family, the key design  \nchoice is the normalisation of the graph operator, since the conversion of affinities into transition probabilities determines which geometric features are preserved. Standard random walks (SRW) normalise row-wise by node degree, causing diffusion to drift toward high-degree, densely sampled region","cbCaie1gf22DNhhD","https://ap.wps.com/l/cbCaie1gf22DNhhD","pdf",14159580,5,1,40,"English","en",105,"# Abstract\n# Introduction\n## Motivation\n## Contributions","[{\"question\":\"What problem does EntroPath address in manifold learning?\",\"answer\":\"EntroPath targets the challenge of recovering intrinsic geodesic geometry from high-dimensional data graphs while avoiding distortions caused by unreliable random-walk normalization or fragile shortest-path estimates.\"},{\"question\":\"How does EntroPath differ from standard diffusion or shortest-path methods?\",\"answer\":\"EntroPath builds dissimilarities from maximum entropy random walks that aggregate the full ensemble of k-step paths, rather than using locally normalized diffusion along a single operator regime or relying on a single shortest-path trajectory.\"},{\"question\":\"What guarantees or theoretical results does the method provide?\",\"answer\":\"The free-energy dissimilarity converges to squared geodesic distance in the short-time limit, established through Varadhan’s heat-kernel formula.\"}]",1784193096,101,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"entropath-maximum-entropy-path-ensemble-embedding-for-manifold-learning","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/entropath-maximum-entropy-path-ensemble-embedding-for-manifold-learning/84122/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does EntroPath address in manifold learning?","Question",{"text":76,"@type":77},"EntroPath targets the challenge of recovering intrinsic geodesic geometry from high-dimensional data graphs while avoiding distortions caused by unreliable random-walk normalization or fragile shortest-path estimates.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does EntroPath differ from standard diffusion or shortest-path methods?",{"text":81,"@type":77},"EntroPath builds dissimilarities from maximum entropy random walks that aggregate the full ensemble of k-step paths, rather than using locally normalized diffusion along a single operator regime or relying on a single shortest-path trajectory.",{"name":83,"@type":74,"acceptedAnswer":84},"What guarantees or theoretical results does the method provide?",{"text":85,"@type":77},"The free-energy dissimilarity converges to squared geodesic distance in the short-time limit, established through Varadhan’s heat-kernel 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