[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81609-en":3,"doc-seo-81609-105":30,"detail-sidebar-cat-0-en-105":91},{"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},81609,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Learning Lineage-guided Geodesics with Finsler Geometry","Trajectory inference reconstructs continuous paths between observed timepoints to estimate system behavior at unobserved times. Prior approaches often rely on continuous geometric priors by defining data-dependent Riemannian metrics. Many real problems also provide discrete, directed transition knowledge—such as lineage trees in developmental biology. This work introduces a Finsler metric that integrates classification signals with geometric structure, combining both continuous and directed priors to improve interpolation performance on synthetic and real data.","Learning Lineage-guided Geodesics with Finsler Geometry  \nAaron Zweig* 1,2 Mingxuan Zhang*2,3 David A. Knowles 1,3,4 Elham Azizi2,3,4,5,6  \n1New York Genome Center  \n2Irving Institute for Cancer Dynamics, Columbia University  \n3Department of Systems Biology, Columbia University  \n4Department of Computer Science, Columbia University  \n5Department of Biomedical Engineering, Columbia University  \n6Data Science Institute, Columbia University  \narXiv :2603 . 16708v2 [ cs .LG] 10 Jul 2026  \nAbstract  \nTrajectory inference investigates how to interpolate paths between observed timepoints of dynamical systems, such as temporally resolved population distributions, with the goal of inferring trajectories at unseen times and better understanding system dynamics. Previous work has focused on continuous geometric priors, utilizing data-dependent spatial features to define a Riemannian metric. In many applications, there exists discrete, directed prior knowledge over admissible transitions (e.g.  \nlineage trees in developmental biology) . We introduce a Finsler metric that combines geometry with classification and incorporate both types of priors in trajectory inference, yielding improved performance on interpolation tasks in synthetic and real-world data.  \n1 INTRODUCTION  \nIn many scientific domains, data are observed at discrete timepoints while the underlying system evolves continuously in time. Trajectory inference aims to reconstruct continuous paths between empirical distributions, enabling interpolation to unseen times and analysis of system evolution. Single-cell omics provides a prominent instance of this setting. Although single-cell RNA sequencing is destructive, sampling at multiple timepoints enables a weakly temporal perspective on the evolution of cells, particularly in developmental settings where cell differentiation is similar among all healthy embryos. However, the inability to sample continuously means a practitioner typically only has access to  \n*  \nThese authors contributed equally.  \nthe cell distribution at a small number of timepoints.  \nIn the context of trajectory inference, optimal transport (OT) is a popular modeling framework across applications because it follows the principle of minimal energy, where the inferred trajectories are optimal with respect to some underlying cost function. In the context of Riemannian metrics, these trajectories are geodesics. However, only in very special cases (Euclidean space or hyperspheres) are the geodesics available in closed form, and must otherwise be learned.  \nWe are interested in settings where the underlying metric is data-dependent, and informed by prior domain knowledge. Encouraging trajectories to stay near observed samples (e.g. cells) [Kapusniak et al., 2024] provides a geometric prior, but one may also include a discrete prior in the form of directed constraints over admissible transitions (e.g. lineage information) . For example, if the literature on developmental cell states in a particular organism demonstrates that one state typically differentiates into another, we seek to enforce that knowledge in our metric to encourage biologically plausible trajectories. Crucially, such priors arise when transitions are known to be structured, directed, or partially ordered (e.g., stage progressions, causal precedence, or permitted state changes), and cannot be captured by symmetric (Riemannian) distances alone.  \nIn this work, we apply Finsler geometry to incorporate discrete, directed transition priors into the geometry used for trajectory inference. Single-cell developmental lineage trees serve as a practical example of this setting in real single-cell RNA sequencing data. Specifically our contributions include (i) the definition of a Finsler metric conditioned on a directed adjacency matrix representing admissible transitions (lineage tree prior),(ii) formal proof that this metric induceswell defined local geometry structure and enforces trajectories that agree wit","cbCait9bfHRkhfc1","https://ap.wps.com/l/cbCait9bfHRkhfc1","pdf",1282174,4,1,12,"English","en",105,"# Introduction\n# Preliminaries\n## Notation\n## Riemannian Metric\n## Finsler Metric","[{\"question\":\"What problem does trajectory inference address in dynamical systems?\",\"answer\":\"Trajectory inference interpolates continuous paths between observed timepoints and infers trajectories at unseen times to better understand system dynamics.\"},{\"question\":\"Why are Riemannian metrics insufficient when prior knowledge is discrete and directed?\",\"answer\":\"Riemannian distances are symmetric and cannot represent structured, directed, or partially ordered transition constraints such as stage progressions in lineage.\"},{\"question\":\"How does the proposed Finsler metric incorporate lineage information?\",\"answer\":\"It defines a Finsler metric conditioned on a directed adjacency matrix of admissible transitions, acting as a penalty when candidate directions contradict the classification signal, thereby enforcing trajectories consistent with the 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problem does trajectory inference address in dynamical systems?","Question",{"text":75,"@type":76},"Trajectory inference interpolates continuous paths between observed timepoints and infers trajectories at unseen times to better understand system dynamics.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are Riemannian metrics insufficient when prior knowledge is discrete and directed?",{"text":80,"@type":76},"Riemannian distances are symmetric and cannot represent structured, directed, or partially ordered transition constraints such as stage progressions in lineage.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed Finsler metric incorporate lineage information?",{"text":84,"@type":76},"It defines a Finsler metric conditioned on a directed adjacency matrix of admissible transitions, acting as a penalty when candidate directions contradict the classification signal, thereby enforcing trajectories consistent with the 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