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arxiv: 2605.23434 · v1 · pith:HN4RIII5new · submitted 2026-05-22 · 💻 cs.LG

Onsager-Machlup Posterior Transport for Deep Gaussian Processes

Jian Xu , Delu Zeng , John Paisley , Qibin Zhao This is my paper

Pith reviewed 2026-05-25 04:46 UTC · model grok-4.3

classification 💻 cs.LG
keywords deep Gaussian processesOnsager-Machlup actionposterior transportDoob bridgevariational inferenceapproximate inferenceUCI regressionprobability-flow ODE
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The pith

Deep Gaussian process inference is recast as learning a deterministic transport map from a reference measure to inducing variables, regularized by the Onsager-Machlup action on a Doob-bridged diffusion.

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

The paper establishes that approximate inference in deep Gaussian processes, normally handled by fitting an explicit variational density over inducing variables, can instead be solved by transporting samples from a tractable reference to the target posterior while penalizing deviation from a path prior. The proposed OM-Path method applies the probability-flow ODE to a Doob-bridged forward SDE and uses the Onsager-Machlup action as the regularizer, yielding an objective that at finite noise equals the negative log density of a tempered path posterior. Theorem 1 further identifies this objective with the small-noise MAP path of the same posterior via the Freidlin-Wentzell large-deviation principle. On seven UCI regression benchmarks the resulting sampler produces statistically significant gains over DBVI on the two largest datasets while the exact-density ELBO ablations do not.

Core claim

OM-Path learns a deterministic sampler that maps a reference measure to posterior-relevant inducing variables by applying Song's probability-flow ODE to DBVI's Doob-bridged forward SDE and regularizing with the Onsager-Machlup action. At the finite noise level used in training the objective is the negative log unnormalised density of a tempered Doob-bridge path posterior; Theorem 1 equates it, via the Freidlin-Wentzell large-deviation principle, to the small-noise MAP path of that posterior. On the UCI suite this produces statistically significant wins over DBVI on the power and protein datasets and ties on two others, while strict path-space ELBO variants on the identical bridge backbone do

What carries the argument

The Onsager-Machlup action applied to the probability-flow ODE of the Doob-bridged reference diffusion, which regularizes the learned transport map and supplies the path-space objective.

If this is right

  • OM-Path records statistically significant NLL and RMSE improvements over DBVI on the two largest UCI datasets under matched-seed paired Wilcoxon tests.
  • The method ties DBVI on yacht and qsar while conceding the three smallest noisy datasets to DBVI.
  • Strict path-space ELBO variants (FFJORD log-det and OM-regularised CNF) on the same bridge backbone fail to beat DBVI on any UCI metric.
  • In this regime, lowering the variance of the path objective matters more than exact density tracking.

Where Pith is reading between the lines

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

  • The transport framing could be applied to other inducing-variable models that already use Doob bridges or score-based diffusion.
  • If the finite-epsilon to small-noise link holds, the same regularizer might be inserted into other variational schemes to reduce estimator variance without changing the target posterior.
  • The observed wins on larger N suggest the method's advantage grows with the dimensionality or complexity of the inducing-variable posterior.

Load-bearing premise

The Freidlin-Wentzell large-deviation principle identification continues to justify the finite-epsilon training objective as a useful approximation to the small-noise MAP path at the noise levels used in practice.

What would settle it

Direct numerical comparison, on a simple DGP, of the paths produced by the finite-epsilon OM-Path objective against the true small-noise MAP paths of the Doob-bridge path posterior would show whether the large-deviation link holds at practical noise values.

Figures

Figures reproduced from arXiv: 2605.23434 by Delu Zeng, Jian Xu, John Paisley, Qibin Zhao.

Figure 1
Figure 1. Figure 1: Bridge marginal coefficients ϕ(s) (mean attenuation) and κ(s) (variance) from the closed￾form ODE system in Prop. 1, for three diffusion strengths λ. With our default λ = g = σ0 = 1 we obtain ϕ(1)≈0.37 and κ(1)≈0.50, both of which appear in the reference drift equation 6 and in the bridge initial distribution sampled in line 4 of Algorithm 1. For posterior transport the situation is reversed: the unnormali… view at source ↗
Figure 2
Figure 2. Figure 2: Per-dataset test RMSE at L = 2 (mean±std, 10 seeds). Bars exceeding 1.5 are clipped and the actual value annotated above. FBVI-bridge-Path and DBVI are jointly the strongest meth￾ods on the small/medium UCI datasets, with the per-dataset winner alternating between them; see Appendix Q for the matched-seed Wilcoxon verdict. 10 20 30 40 50 60 70 80 90 100 Epoch 1.05 1.10 1.15 1.20 1.25 1.30 1.35 Test NLL Tes… view at source ↗
Figure 3
Figure 3. Figure 3: Test-NLL trajectories on protein (L = 2). Solid lines are seed-mean; shaded bands are ±1 std. FBVI-bridge-Path (OM, our main method, 10 seeds) and FBVI-bridge (implicit-q, 3 seeds re￾run under matched protocol) converge to NLL ≈1.10 by epoch 100, with the OM variant nominally lowest (1.086±0.034); DSVI plateaus higher at ≈1.16; unbridged FBVI has a single-seed instability event near epoch 68. SGHMC / IPVI … view at source ↗
Figure 4
Figure 4. Figure 4: Depth-scaling test RMSE on the seven small/medium UCI datasets (mean over 10 seeds; [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Few-step inference: test RMSE as a function of Euler integration steps at evaluation time [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Simple regret vs. BO iteration on four synthetic black-box functions, mean [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
read the original abstract

Approximate inference over inducing variables is the central computational bottleneck of Deep Gaussian Processes (DGPs). Existing methods either fit an explicit density $q_\phi(\bU)$ by an ELBO (DSVI, IPVI, DDVI, DBVI) or sample by MCMC (SGHMC). We instead frame DGP inference as \emph{posterior transport}: learn a deterministic sampler that maps a tractable reference measure to posterior-relevant inducing variables, regularised by a path prior derived from the Doob-bridged reference diffusion. Our realisation, \textbf{OM-Path} (formally FBVI-bridge-Path), uses Song's probability-flow ODE applied to DBVI's Doob-bridged forward SDE; the reference drift is closed-form from the bridge marginal coefficients (no score matching) and the path regulariser is the \textbf{Onsager--Machlup action}. At the finite-$\epsilon$ value used at training, the objective is the negative log unnormalised density of a tempered Doob-bridge path posterior, and Theorem 1 identifies it with the same posterior's small-noise MAP path via the Freidlin--Wentzell LDP. Two strict path-space ELBO variants on the same bridge backbone (FFJORD log-det; OM-regularised CNF) are derived as ablations. Under a matched-seed paired Wilcoxon test against DBVI on seven UCI regression benchmarks, OM-Path delivers statistically significant wins on the two largest datasets (\textit{power}: $p\!=\!0.014$, NLL $\mathbf{0.012}$ matching the DSVI baseline of $0.017$; \textit{protein}: $p\!=\!0.002$, RMSE $\mathbf{0.716}$ vs.\ $0.764$, NLL $\mathbf{1.086}$ vs.\ $1.149$), statistical ties on \textit{yacht} / \textit{qsar}, and concedes \textit{boston} / \textit{energy} / \textit{concrete} to DBVI on small-$N$ noisy data. The strict-ELBO variants do not clear DBVI on any UCI metric: in this regime, reducing the variance of the path objective dominates exact-density tracking.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript frames DGP inducing-variable inference as posterior transport and introduces OM-Path (FBVI-bridge-Path), which applies Song's probability-flow ODE to DBVI's Doob-bridged forward SDE with the Onsager-Machlup action as path regularizer. At the finite ε used in training the objective is the negative log unnormalised density of a tempered Doob-bridge path posterior; Theorem 1 identifies this quantity with the same posterior's small-noise MAP path via the Freidlin-Wentzell LDP. On seven UCI regression benchmarks a matched-seed paired Wilcoxon test against DBVI reports statistically significant wins on the two largest datasets (power, protein), ties on yacht/qsar, and losses on the three smallest noisy sets. Strict path-space ELBO ablations (FFJORD log-det and OM-regularised CNF) on the same bridge backbone do not beat DBVI.

Significance. If the LDP identification holds at the operating ε, the work supplies a new, closed-form-drift path-regularisation route for DGP inference that can improve upon DBVI on larger data while avoiding score matching. The explicit derivation of two strict path-space ELBO variants as ablations and the reporting of both wins and losses across dataset sizes are positive features. The empirical gains remain modest and dataset-size dependent, so the result is of moderate rather than transformative significance even if the theory is confirmed.

major comments (2)
  1. [Theorem 1] Theorem 1: the identification of the finite-ε Onsager-Machlup training objective with the small-noise MAP path rests on the Freidlin-Wentzell LDP. The manuscript must supply either an explicit bound on ε or a numerical verification that the LDP regime is attained at the noise levels actually used during training; without this check the claimed equivalence between the path regulariser and the posterior-transport interpretation is not yet supported.
  2. [Experimental results] § on experimental results (UCI tables): the paired Wilcoxon tests are reported only for the two largest datasets; the manuscript should state whether a multiple-comparison correction was applied across the seven datasets and whether the reported p-values remain significant after correction, because the central empirical claim of superiority over DBVI is carried by these two wins.
minor comments (2)
  1. [Method] The abstract states that the reference drift is closed-form from the bridge marginal coefficients; the corresponding derivation should be placed in the main text rather than left implicit.
  2. [Preliminaries] Notation for the tempered Doob-bridge path posterior and the precise definition of the Onsager-Machlup action at finite ε should be introduced once and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments below.

read point-by-point responses

  1. Referee: [Theorem 1] Theorem 1: the identification of the finite-ε Onsager-Machlup training objective with the small-noise MAP path rests on the Freidlin-Wentzell LDP. The manuscript must supply either an explicit bound on ε or a numerical verification that the LDP regime is attained at the noise levels actually used during training; without this check the claimed equivalence between the path regulariser and the posterior-transport interpretation is not yet supported.

    Authors: Theorem 1 establishes the identification strictly in the small-noise limit via the Freidlin-Wentzell LDP; at finite ε the objective is an approximation to the MAP path. We will revise the manuscript to include a numerical check on a low-dimensional synthetic example, comparing OM-regularised paths at the training ε values against directly optimised small-noise MAP paths, to confirm the operating regime is consistent with the LDP approximation. revision: yes

  2. Referee: [Experimental results] § on experimental results (UCI tables): the paired Wilcoxon tests are reported only for the two largest datasets; the manuscript should state whether a multiple-comparison correction was applied across the seven datasets and whether the reported p-values remain significant after correction, because the central empirical claim of superiority over DBVI is carried by these two wins.

    Authors: Each UCI dataset constitutes an independent experimental condition with distinct data characteristics and noise levels; the tests therefore evaluate dataset-specific behaviour rather than repeated tests of a single global hypothesis. No multiple-comparison correction was applied. The manuscript already reports the full pattern of wins on the two largest sets, ties on two others, and losses on the three smallest noisy sets. In revision we will add the p-values for all seven datasets for completeness while retaining the per-dataset interpretation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external LDP and independent empirical benchmarks

full rationale

The paper's core construction applies Song's probability-flow ODE to an existing DBVI Doob bridge, adds an Onsager-Machlup path regularizer, and states that the finite-ε objective equals the negative log unnormalised tempered Doob-bridge path posterior. Theorem 1 then invokes the standard external Freidlin-Wentzell large-deviation principle to identify this with the small-noise MAP path. Neither step reduces the claimed result to a fitted parameter, a self-citation loop, or a definitional tautology. The UCI regression results (matched-seed Wilcoxon tests) supply an independent external benchmark. No load-bearing self-citation, ansatz smuggling, or renaming of known results appears in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger extracted from abstract only; full paper may contain additional fitted quantities or domain assumptions.

axioms (2)
  • standard math Freidlin-Wentzell large deviation principle applies to the Doob-bridged diffusion paths at the finite noise level used in training
    Invoked by Theorem 1 to equate the tempered path-posterior objective with the small-noise MAP path.
  • domain assumption Reference drift of the probability-flow ODE is closed-form from the bridge marginal coefficients without requiring score matching
    Stated as a property that removes the need for score estimation.

pith-pipeline@v0.9.0 · 5958 in / 1656 out tokens · 36264 ms · 2026-05-25T04:46:25.279752+00:00 · methodology

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Reference graph

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