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

2026-04-22

AI Summary

Trajectory inference suffers from non-identifiability of path distributions in function space due to destructive measurements, and existing evaluation methods are limited to marginal predictions and yield inconsistent results. This work proposes the first general framework capable of computing Kullback–Leibler (KL) divergence directly in function space, enabling consistent, path-level assessment of trajectory inference methods under partial observability. The approach leverages relative entropy estimation between probability measures over function spaces and introduces a scalable, data-driven estimator. In benchmark experiments, the method shows excellent agreement with analytically computed KL divergences and effectively discriminates performance differences among trajectory inference algorithms on both real and synthetic single-cell RNA sequencing datasets, substantially improving evaluation reliability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

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.

Problem

• Trajectory Inference
• KL divergence
• function space
• snapshot data
• partial observability

Innovation

• functional KL divergence
• trajectory inference
• path-space evaluation
• snapshot data
• relative entropy estimation