Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inference

• URL: http://arxiv.org/abs/2604.20775v1
• Date: Wed, 22 Apr 2026 17:03:49 GMT
• Title: Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inference
• Abstract summary: Trajectory Inference (TI) seeks to recover latent dynamical processes from snapshot data.<n>In applications such as single-cell genomics, destructive measurements make path-space laws non-identifiable from finitely many marginals.<n>We introduce a general framework for estimating the Kullback-Leibler divergence (KL) divergence between probability measures on function space.
• Score: 12.968551579672214
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Trajectory Inference (TI) seeks to recover latent dynamical processes from snapshot data, where only independent samples from time-indexed marginals are observed. In applications such as single-cell genomics, destructive measurements make path-space laws non-identifiable from finitely many marginals, leaving held-out marginal prediction as the dominant but limited evaluation protocol. We introduce a general framework for estimating the Kullback-Leibler divergence (KL) divergence between probability measures on function space, yielding a tractable, data-driven estimator that is scalable to realistic snapshot datasets. We validate the accuracy of our estimator on a benchmark suite, where the estimated functional KL closely matches the analytic KL. Applying this framework to synthetic and real scRNA-seq datasets, we show that current evaluation metrics often give inconsistent assessments, whereas path-space KL enables a coherent comparison of trajectory inference methods and exposes discrepancies in inferred dynamics, especially in regions with sparse or missing data. These results support functional KL as a principled criterion for evaluating trajectory inference under partial observability.

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