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Learning Lineage-guided Geodesics with Finsler Geometry

Aaron Zweig, Mingxuan Zhang, David A. Knowles, Elham Azizi

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.

Learning Lineage-guided Geodesics with Finsler Geometry

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.
Paper Structure (22 sections, 3 theorems, 25 equations, 4 figures, 5 tables, 1 algorithm)

This paper contains 22 sections, 3 theorems, 25 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Riemannian Metric
  5. Finsler Metric
  6. Related Work
  7. Trajectory Inference Methods
  8. Cell Lineage Tree Priors
  9. Riemannian and Finsler Metrics
  10. Lineage-Informed Finsler Geometry
  11. Lineage Tree Guided Finsler Metric
  12. Parameterizing Finsler Geometry
  13. Learning Finsler Geometry
  14. Results
  15. Synthetic Data
  16. ...and 7 more sections

Key Result

Proposition 4.1

Given that $\|v\|_{g(x)}$ is a non-trivial conformal Riemannian metric, $F$ defines a non-degenerate Finsler metric.

Figures (4)

  • Figure 1: A visual representation of the Finsler metric as a penalty. For temporal data evolving from $t=0$ (red) towards $t=1$ (blue), we want to define a local metric at $x$ such that, if $v_1$ agrees with the lineage prior and $v_2$ contradicts the prior, then $F(x, v_1) < F(x, v_2)$, i.e. the metric acts as a penalty on disagreeing with the classification signal.
  • Figure 2: The Finsler metric trained on synthetic 2d data. The true lineage goes from class $0 \rightarrow 1 \rightarrow 4$, but the model is only trained on $t \in \{0, 2\}$ and therefore must rely on classification guidance to bias the trajectories.
  • Figure 3: Cell type predictions along two learned trajectories for the Mouse-Blood dataset. The x-axis is time and the y-axis is cell type probability. Top and bottom are two sampled trajectories.
  • Figure 4: Cell type predictions along five learned trajectories for the Zebrafish-CNS dataset. The x-axis is time and the y-axis is cell type probability. Top and bottom are five sampled trajectories.

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Proposition 4.1
  • proof
  • Theorem 4.2: Local Finsler structure
  • proof
  • Theorem 4.3: Geodesic recovery and non-contradicting path
  • proof