Learning Lineage-guided Geodesics with Finsler Geometry

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

cs.LGMar 17, 2026

Learning Lineage-guided Geodesics with Finsler Geometry

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

Organizations: New York Genome Center · Irving Institute for Cancer Dynamics, Columbia University · Department of Systems Biology, Columbia University · Department of Computer Science, Columbia University ·

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.

Figures & tables

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.

Panel of Figure 2 — 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 through the dark blue cell type.

	Synthetic( $n$ =2)	Synthetic( $n$ =50)
CFM( Tong et al. [2024a] )	$0.95\pm 0.00$	$1.21\pm 0.00$
CFM+Finsler	$\bm{0.43\pm 0.02}$	$\bm{0.97\pm 0.03}$

Table 1: Wasserstein-1 distance averaged over marginals on left-out time points for the synthetic dataset.

n

represents feature dimension.

	Zebrafish-CNS	Zebrafish-PA
LineageOT( Forrow and Schiebinger [2021] )	$15.01\pm 0.00$	$\bm{11.68\pm 0.00}$
Moslin( Lange et al. [2024] )	$14.67\pm 0.00$	$12.15\pm 0.00$
$\text{SF}^{2}\text{M}$ ( Tong et al. [2024b] )	$15.91\pm 0.11$	$13.42\pm 0.10$
CFM( Tong et al. [2024a] )	$14.69\pm 0.02$	$12.16\pm 0.02$
CFM+Finsler	$\bm{14.57\pm 0.02}$	$11.99\pm 0.04$
MFM( Kapusniak et al. [2024] )	$13.85\pm 0.03$	$\bm{11.42\pm 0.03}$

Table 2: Wasserstein-1 distance averaged over marginals on left-out time points for 100-PC representation of the zebrafish dataset.

Panel of Figure 3 — 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.

	Mouse-Blood
LineageOT( Forrow and Schiebinger [2021] )	$7.98\pm 0.00$
Moslin( Lange et al. [2024] )	$14.67\pm 0.00$
$\text{SF}^{2}\text{M}$ ( Tong et al. [2024b] )	$9.16\pm 0.01$
CFM( Tong et al. [2024a] )	$8.07\pm 0.08$
CFM+Finsler	$\bm{7.65\pm 0.04}$
MFM( Kapusniak et al. [2024] )	$7.95\pm 0.08$

Table 3: Wasserstein-1 distance averaged over marginals on left-out time points for 100-PC representation of the mouse embryo organogenesis dataset.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Range
$\lambda$	$\{0.2,0.5,1.0\}$
classifier smoothing	$\{0.03,0.05\}$

Appendix

Table 4: Hyperparameter ranges for Finsler metric parameters

Hyperparameter	Range
$\#$ clusters $r$	$\{100,150,200\}$
kernel bandwidth $\kappa$	$\{1,1.5\}$
kernel smoothing parameter $\epsilon$	$\{0.05,0.1,0.2\}$
use Euclidean optimal transport	$\{True,False\}$

Appendix

Table 5: Hyperparameter ranges for Metric Flow Matching specific parameters

Panel of Figure 4 — 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.

Panel of Figure 5 — Figure 5: Lineage tree priors used in all single-cell experiments

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Oct 8, 2026J/K move · Enter open · S save

Finsler Flow Matching: Dynamics-Aware Geodesic Interpolation for Single-Snapshot Trajectory Inference

Niklas Canova, Jonas Simon Fleck

Institute of Human Biology (IHB), Roche, Basel, Switzerland

Finsler Geometry Flow-Based Generative Modeling Single-Cell Transcriptomics Single-Cell RNA Sequencing Conditional Flow Matching

Open paper AskPDF arXiv

Single-cell snapshot data can resolve a continuum of cellular states but do not uniquely determine the dynamics governing transitions between them. However, additional dynamical information can often be encoded in a cell-cell Markov transition kernel. Existing generative approaches for single cell trajectory inference either infer transport only from population marginals, impose a symmetric geometry on the state space, or incorporate directionality through a single velocity vector at each observed state. We introduce Finsler Flow Matching (FFM), a framework for learning continuous stochastic dynamics from discrete Markov transition graphs. We use the first and second local moments to construct a Finsler structure motivated by the Freidlin--Wentzell action, where the second moment determines anisotropic accessibility and the first moment introduces a preferred direction of motion. We learn neural approximations of the resulting directed geodesics, use their Finsler cost to construct source-target couplings, and define geometry-aware stochastic conditional paths that can be distilled into a continuous generative process through simulation-free score and flow matching. Across synthetic and single-cell trajectory inference benchmarks, FFM improves recovery of withheld intermediate populations, particularly when the transition dynamics are strongly directional or anisotropic. Our results provide a principled route from discrete transition probabilities to continuous generative dynamics while retaining both directional and diffusive structure.

Abstract

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