Onsager-Machlup Posterior Transport for Deep Gaussian Processes

arXiv:2605.23434v1 Announce Type: new 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(U) by an ELBO (DSVI, IPVI, DDVI, DBVI) or sample by MCMC (SGHMC). We instead frame DGP inference as 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, extbf{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 extbf{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 (extit{power}: p!=!0.014, NLL mathbf{0.012} matching the DSVI baseline of 0.017; extit{protein}: p!=!0.002, RMSE mathbf{0.716} vs. 0.764, NLL mathbf{1.086} vs. 1.149), statistical ties on extit{yacht} / extit{qsar}, and concedes extit{boston} / extit{energy} / extit{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.

Source: arXiv cs.LG | 2026-05-25