Onsager-Machlup Posterior Transport for Deep Gaussian Processes
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| Format: | Preprint |
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2026
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| _version_ | 1866918518226681856 |
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| author | Xu, Jian Zeng, Delu Paisley, John Zhao, Qibin |
| author_facet | Xu, Jian Zeng, Delu Paisley, John Zhao, Qibin |
| contents | Approximate inference over inducing variables is the central computational bottleneck of Deep Gaussian Processes (DGPs). Existing methods either fit an explicit density $q_ϕ(\bU)$ 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-$ε$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23434 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Onsager-Machlup Posterior Transport for Deep Gaussian Processes Xu, Jian Zeng, Delu Paisley, John Zhao, Qibin Machine Learning Approximate inference over inducing variables is the central computational bottleneck of Deep Gaussian Processes (DGPs). Existing methods either fit an explicit density $q_ϕ(\bU)$ 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-$ε$ 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. |
| title | Onsager-Machlup Posterior Transport for Deep Gaussian Processes |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2605.23434 |