Learning Lineage-guided Geodesics with Finsler Geometry

2026-03-17

AI Summary

This work addresses the challenge of integrating directional priors with continuous geometric information in trajectory inference for dynamic systems. To this end, we propose a novel approach grounded in Finsler geometry, introducing— for the first time in this context—a Finsler metric that encodes direction-sensitive, asymmetric distances. This framework unifies discrete, directed lineage priors with data-driven continuous spatial geometry, thereby guiding geodesic learning in a manner that respects inherent system directionality. By jointly leveraging trajectory interpolation, lineage constraints, and geodesic optimization, our method achieves substantially improved interpolation accuracy on both synthetic and real biological datasets, enabling more faithful reconstruction of system dynamics at unobserved time points. This advance overcomes a key limitation of conventional Riemannian frameworks, which inherently neglect directional asymmetry.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Trajectory 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. lineage 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.

Problem

• trajectory inference
• Finsler geometry
• lineage prior
• geodesic interpolation
• dynamical systems

Innovation

• Finsler geometry
• trajectory inference
• lineage priors
• geodesic interpolation
• Riemannian metric