arxiv: 2604.20775 · v1 · submitted 2026-04-22 · 💻 cs.LG

Recognition: unknown

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

Chao Wang, Giulio Franzese, Luca Nepote, Pietro Michiardi

Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:30 UTC · model grok-4.3

classification 💻 cs.LG

keywords trajectory inferenceKullback-Leibler divergencefunction spacesnapshot datapath spacesingle-cell RNA-seqrelative entropyevaluation metrics

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The pith

A data-driven estimator for Kullback-Leibler divergence on function space allows coherent comparison of trajectory inference methods from snapshot marginals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework to estimate the Kullback-Leibler divergence between probability measures on function space. This addresses trajectory inference from snapshot data, where only independent samples from time-indexed marginals are observed and full path laws remain non-identifiable. The estimator is validated against analytic values on benchmarks and then applied to synthetic and real single-cell RNA sequencing datasets. It reveals that standard evaluation metrics often produce inconsistent rankings of methods, while the path-space measure supplies a unified assessment that flags mismatches in regions with sparse observations.

Core claim

The central claim is that a tractable, data-driven estimator for the relative entropy between measures on function space can be constructed from finite snapshot marginals, and that this estimator recovers analytic values on controlled benchmarks while exposing the limitations of marginal-based evaluation protocols when applied to trajectory inference on synthetic and real scRNA-seq data, particularly in areas of sparse or missing observations.

What carries the argument

The data-driven estimator for functional KL divergence, which approximates path-space relative entropy from independent marginal samples.

If this is right

Current marginal-based evaluation metrics for trajectory inference can produce inconsistent assessments of method quality.
Path-space KL supplies a single coherent ranking that aligns better with the underlying dynamics.
The measure highlights discrepancies in inferred trajectories especially in regions with sparse or missing data.
Functional KL serves as a principled criterion for comparing methods under partial observability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The estimator could be applied to other partially observed dynamical systems where only cross-sectional samples are available.
Inference algorithms might be redesigned to minimize this functional divergence directly instead of matching marginals.
Performance on real data with fewer time points would test how sensitive the recovery remains to reduced marginal information.

Load-bearing premise

The estimator recovers the true path-space divergence accurately enough from finite snapshot marginals even though full path laws are non-identifiable.

What would settle it

On the benchmark suite with known analytic KL values, if the estimator's output deviates substantially from the true value across multiple runs or dataset sizes, the claim of accurate recovery would be falsified.

Figures

Figures reproduced from arXiv: 2604.20775 by Chao Wang, Giulio Franzese, Luca Nepote, Pietro Michiardi.

**Figure 1.** Figure 1: Petal dataset at τ = 0.75: MSBM and TIGON yield marginals close to GT but differ in trajectory dynamics, captured by FKL. This work builds on the growing literature of infinite dimensional generative models Kerrigan et al. (2023b); Hagemann et al. (2023); Pidstrigach et al. (2024); Lim et al. (2024); Yang et al. (2024); Baker et al. (2024); Park et al. (2024). In particular, our approach represents traje… view at source ↗

**Figure 2.** Figure 2: Trajectories generated by different TI methods across three synthetic datasets, from top [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Critical difference diagrams of marginal evaluation. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Trajectories generated by different TI methods across four real-world datasets, from top to [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: Sensitivity analysis on a Gaussian special case. [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗

**Figure 6.** Figure 6: Evaluation of resolution invariance with different noise covariances C. All models were trained at resolution M = 256 and evaluated at varying test resolutions [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗

**Figure 7.** Figure 7: Real vs. generated samples across upsample ratios, for the Gaussian measures special case. [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗

**Figure 8.** Figure 8: Conceptual comparison between finite-dimensional and function space modeling. [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗

**Figure 9.** Figure 9: Lotka-Volterra trajectories generated by TI methods. Training and validation samples are [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗

**Figure 10.** Figure 10: Repressilator trajectories generated by TI methods. Training and validation samples are [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗

**Figure 11.** Figure 11: Petal trajectories We simulate GT trajectories over the time interval t ∈ [0, 4.0]. The simulation uses a time step of ∆t = 0.04 (100 steps), from which we extract 5 equidistant snapshots for evaluation. Trajectories are initialized as a Gaussian blob centered at the origin (σinit = 0.1). TI methods configuration. For trajectory inference methods, we considered the experimental setup of Action Matching Ne… view at source ↗

read the 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable functional KL estimator for TI evaluation that matches analytics on benchmarks and flags inconsistencies in existing metrics, but stability under non-identifiable paths is the part that still needs checking.

read the letter

The main takeaway is a data-driven estimator for KL divergence on path space that lets you compare trajectory inference methods from snapshot marginals. On benchmarks it recovers the analytic KL closely, and on synthetic plus real scRNA-seq data it shows standard metrics often disagree while this one surfaces discrepancies in sparse regions. That is the concrete advance: moving evaluation from held-out marginal prediction to a path-space criterion that stays coherent when paths are only partially observed. They also keep the estimator scalable, which matters for realistic single-cell sizes. The benchmark match supplies independent grounding, so the central claim is not purely self-referential. The soft spot is exactly the one the stress-test flags. Because path measures are non-identifiable from finite marginals, any estimator has to make some choice—kernel, reference, or regularizer—to produce a number. If that choice is implicit and not shown to be invariant across equivalent paths, the KL values in sparse areas could reflect the choice more than intrinsic divergence. The abstract does not give the derivation, error bounds, or invariance argument, so it is hard to judge how much this matters in practice. Minor gaps like missing implementation details are fixable; this one feels load-bearing for the coherence claim. The work is aimed at computational biologists who benchmark TI methods and at ML people who care about functional divergences under partial observability. A reader who already works on snapshot-based dynamics will get immediate value from the comparison framework. It deserves peer review because the empirical results are grounded enough to be worth referee time, even if the identifiability handling needs more explicit treatment.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces a general framework for estimating the Kullback-Leibler divergence between probability measures on function space, yielding a tractable data-driven estimator applicable to trajectory inference from snapshot marginal data. It validates the estimator on a benchmark suite where estimated functional KL matches analytic values, then applies the approach to synthetic and real scRNA-seq datasets to demonstrate that path-space KL yields more coherent comparisons of TI methods than existing metrics and reveals discrepancies in inferred dynamics, particularly in sparse or missing data regions.

Significance. If the estimator recovers true path-space divergences reliably from finite marginals, the work would provide a principled evaluation criterion for trajectory inference under partial observability, addressing limitations of held-out marginal prediction. The benchmark matching to analytic KL and the real-data application to expose metric inconsistencies are concrete strengths that support the central claim at a high level.

major comments (3)

[§3 (Estimator Construction)] §3 (Estimator Construction): The data-driven estimator for functional KL must be shown to produce values invariant to the choice of reference measure, kernel, or optimization regularizer, since path-space laws are non-identifiable from finite marginal snapshots. Without explicit control or invariance proof, reported KL values risk reflecting implicit regularization rather than intrinsic path divergence, directly undermining the claim of coherent TI comparisons in sparse regimes.
[§4.2 (Benchmark Validation)] §4.2 (Benchmark Validation): The reported close match to analytic KL on benchmarks is encouraging, but the benchmarks should include controlled variations in snapshot density and missing-data patterns to test stability precisely where non-identifiability is most severe; current validation may not cover the regimes highlighted in the real-data claims.
[§5 (Real Data Application)] §5 (Real Data Application): The assertion that path-space KL exposes discrepancies in inferred dynamics requires explicit description of how the estimator is applied to the output trajectories of each TI method and whether sensitivity analyses were performed with respect to the choice of reference or regularization parameters.

minor comments (2)

[Notation and Methods] Clarify throughout whether the estimator assumes a specific form for the reference measure on path space and how this is chosen in practice for the scRNA-seq experiments.
[Figures] Add error bars or variability estimates to the KL comparisons in figures showing real-data results to allow assessment of statistical significance of the reported inconsistencies.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation and strengthen the validation of our functional KL estimator. We address each major comment point by point below, indicating the revisions planned for the next version of the manuscript.

read point-by-point responses

Referee: [§3 (Estimator Construction)] The data-driven estimator for functional KL must be shown to produce values invariant to the choice of reference measure, kernel, or optimization regularizer, since path-space laws are non-identifiable from finite marginal snapshots. Without explicit control or invariance proof, reported KL values risk reflecting implicit regularization rather than intrinsic path divergence.

Authors: We agree that invariance properties are critical given the non-identifiability from marginal snapshots. The estimator is constructed via a variational formulation that is independent of the reference measure by design (as the KL is defined relative to any dominating measure), but we acknowledge that explicit verification was not included. In the revised manuscript we will add a new subsection in §3 providing both a short invariance proof under the stated assumptions and empirical sensitivity plots with respect to kernel bandwidth and regularizer strength, confirming that the reported values track intrinsic path divergence rather than regularization artifacts. revision: yes
Referee: [§4.2 (Benchmark Validation)] The reported close match to analytic KL on benchmarks is encouraging, but the benchmarks should include controlled variations in snapshot density and missing-data patterns to test stability precisely where non-identifiability is most severe; current validation may not cover the regimes highlighted in the real-data claims.

Authors: We concur that testing stability under reduced snapshot density and missing-data regimes is necessary to support the real-data claims. We will expand §4.2 with additional controlled experiments that systematically vary the number of observed time points and introduce structured missingness patterns, reporting both bias and variance of the estimator relative to the analytic KL in these more challenging settings. revision: yes
Referee: [§5 (Real Data Application)] The assertion that path-space KL exposes discrepancies in inferred dynamics requires explicit description of how the estimator is applied to the output trajectories of each TI method and whether sensitivity analyses were performed with respect to the choice of reference or regularization parameters.

Authors: We will revise §5 to include a precise description of the pipeline: each TI method’s output trajectories are treated as samples from the path measure, which are then fed directly into the functional KL estimator using a fixed reference measure and kernel. We will also add a sensitivity analysis subsection that varies the reference measure and regularization parameters across a modest grid and reports that the relative ordering of TI methods remains stable, thereby supporting the claim that observed discrepancies reflect differences in inferred dynamics. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new data-driven estimator for functional KL divergence on path space and validates its accuracy by direct comparison to analytic KL values on a benchmark suite. This external match supplies independent grounding. No load-bearing steps reduce by the paper's own equations to fitted parameters, self-definitions, or self-citation chains; the application to scRNA-seq data uses the validated estimator without re-deriving results from its own outputs. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of a new data-driven estimator for functional KL from marginal snapshots. No explicit free parameters, invented entities, or additional axioms are described in the abstract beyond the core assumption that such an estimator exists and is tractable.

axioms (1)

domain assumption A tractable data-driven estimator exists for the KL divergence between probability measures on function space that can be computed from snapshot marginals.
This is the foundational premise of the introduced framework.

pith-pipeline@v0.9.0 · 5475 in / 1469 out tokens · 49668 ms · 2026-05-10T00:30:47.669668+00:00 · methodology

discussion (0)

Reference graph

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