Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inference
Chao Wang, Luca Nepote, Giulio Franzese, Pietro Michiardi
TLDR
This paper introduces a novel framework for estimating Kullback-Leibler divergence in function space, improving trajectory inference evaluation.
Key contributions
- Introduces a general framework for estimating KL divergence between probability measures on function space.
- Develops a tractable, data-driven estimator scalable for realistic snapshot datasets.
- Validates the estimator's accuracy, showing functional KL closely matches analytic KL.
- Demonstrates functional KL provides a coherent evaluation for trajectory inference methods, exposing discrepancies.
Why it matters
Current trajectory inference evaluation methods are often inconsistent and limited. This work provides a principled, data-driven metric (functional KL) that offers a more robust and accurate way to compare and validate TI models, especially in complex biological applications like scRNA-seq. This advances the reliability of dynamic process recovery.
Original 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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