Bridge Graphical Models

1. From latent codes to latent paths

The broad idea behind Bridge Graphical Models is not merely that a velocity predictor has nonzero error. It is that continuous-time generative models contain a latent path graphical model, and a sampler must turn that endpoint-informed latent object into a causal computation.

This hierarchy is important. The path-space VAE interpretation, probability-current gauge, and augmented field geometry belong to the ambient BGM program. The Markovization gap is the first sharply measurable obstruction inside that program; it does not by itself complete the whole unification.

2. What the earlier wording undersold—and oversold

We formulate these models as endpoint-conditioned latent path processes and study the compression from non-causal bridge information to causal sampler representations. The Markovization gap quantifies the target-, observation-, and clock-relative irreducible squared-loss uncertainty of this projection.

Here “non-causal” does not mean a physical violation of causality. It means that the training target may depend on a terminal endpoint or bridge randomness that is unavailable to the sampler at generation time. “Causal sampler representation” means any state constructed from information that the sampler is permitted to use.

3. The Markovization gap after the rebuttal

Let \(L\sim\mathscr Q\) denote the full bridge record: endpoints, bridge randomness, latent or augmented coordinates, and any relevant history. The training objective selects a local target \(U_t=u_t(L)\). The sampler receives only

For a training-time law \(\mu\), define

For squared loss, the Bayes decoder is

and the classical Hilbert-space projection identity gives

If a richer state \(S'_t\) contains at least as much information as \(S_t\), then conditioning reduces variance:

Memory or augmented latent state can therefore reduce, and sometimes eliminate, the gap. It is an information-relative bottleneck, not an impossibility result for every stateful generator.

4. Why a 3D Gaussian experiment is still an \(L_2\) regression problem

The words “3D Gaussian” and “\(L_2\) regression” describe different aspects of the same experiment:

The coupling changes the distribution under which this regression is evaluated. The observation map changes how much of that distribution the decoder can see. Neither changes the fact that the measured floor is squared \(L_2\) error.

The decisive observation control

The same straight Monge bridge keeps endpoint cost and total target variance fixed. Only the sampler-visible state changes.

The value \(4\times10^{-6}\) is not a new nonzero gap; it is numerical/optimization residue in a case whose analytic floor is exactly zero. The substantive result is the transition \(0\to0.227\to0.575\) caused solely by information removal.

Across 11 coupling, bridge, and observation configurations, exact gap versus MLP test MSE has Spearman and Pearson correlation \(1.0000\) to four decimals; learned error is within about 2% of the analytic floor.

5. Coupling changes the regression problem before training

The exact Gaussian experiment fixes the endpoint marginals and gradually degrades their cross-covariance. This turns the original two-point OT-versus-independent comparison into a graded intervention.

For the straight target, the exact floor follows \(0\to0.182\to0.749\to2.270\to7.484\). The same endpoint marginals therefore induce very different Markov-decoder problems depending on how endpoints are paired.

Fashion-MNIST coupling continuum

One time-conditioned decoder architecture, optimizer, training budget, and three seeds are held fixed. Starting from minibatch OT, progressively larger fractions of the assignment are scrambled before reaching independence; Brownian OT changes bridge target at the original OT endpoint cost.

The proxy and validation MSE have Spearman correlation \(1.000\) over all six designs and over the five straight couplings alone. Brownian and straight OT share endpoint cost, yet their validation errors differ sharply, so endpoint cost alone cannot rank bridge targets.

6. What the practical proxy does—and does not estimate

The population conditional variance is generally unavailable on images. The primary practical score is cross-fitted conditional-mean error:

This is a finite-sample ranking proxy. It contains conditional-mean estimation error and is not an unbiased estimator, a certified lower bound, or a number that should be compared across unrelated feature pipelines.

Calibration against known exact gaps

A 32D Gaussian family with known population gaps varies \(N\in\{2000,5000,10000\}\), \(k\in\{16,32,64\}\), PCA dimension \(\{8,16,32\}\), seven bridge/coupling configurations, and three seeds. At \(N=10000\), the median exact-versus-proxy Spearman correlation is \(1.000\).

Cross-fitted error is preferred because it directly asks the operational question—how well can a predictor trained on other samples recover the target at a held-out state? Local variance additionally mixes conditional uncertainty with variation of the conditional mean across a finite neighborhood.

7. Broader bridge families: strong signal, imperfect oracle

The CIFAR-10 experiment holds architecture and optimization budget fixed while varying deterministic schedules, a VP-cosine path, Brownian targets at three noise levels, independent coupling, and a VE corruption control. The seven principal OT-coupled rows have identical endpoint cost \(3738.17\).

Across these seven designs, proxy versus validation MSE has Spearman \(0.893\) and Pearson \(0.897\). Within the Brownian-noise family, both increase monotonically. The fixed endpoint cost rejects transport cost as a complete explanation.

CIFAR-100: lower regression difficulty does not guarantee better samples

Minibatch OT lowers the proxy and training loss, but it does not improve FID, KID, or IS in this pilot. This is a direct boundary case for the claim. The retained pilot summary has no seed-level uncertainty, so it is not presented as a powered benchmark.

8. What lies outside the Markovization layer

The original research vision remains broader than the present theorem. A useful stack is

9. Computational complexity

For \(N_{\mathrm{tr}}\) reference samples, \(N_{\mathrm{te}}\) held-out samples, feature dimension \(r\), and \(M\) sampled times, brute-force cross-fitted \(k\)-NN computes every train–test distance at every time:

Storing all distances would require \(O(N_{\mathrm{tr}}N_{\mathrm{te}})\) memory. Query chunks of size \(b\) reduce working distance memory to \(O(bN_{\mathrm{tr}})\). Projecting \(N\) original \(d\)-dimensional features into \(r\) dimensions costs \(O(Ndr)\) after the PCA basis is fitted.

Coupling construction is separate. Independent pairing is \(O(Nd)\). Exact Hungarian matching within batches of size \(B\) costs \(O(B^3)\) per batch, hence \(O(NB^2)\) over \(N/B\) batches, with \(O(B^2)\) memory. A learned coupling or feature encoder adds its own training and inference cost.

10. What the current evidence supports

The target- and observation-relative Markovization gap is the exact irreducible regression floor in solvable bridge models. Cross-fitted PCA/\(k\)-NN error is an inexpensive, empirically rank-calibrated proxy for absolute fixed-budget Markov-decoder difficulty across the tested coupling and bridge designs. It is not a certified population estimate, a scale-free score, or a universal predictor of final generative quality.

11. Design rules after the rebuttal

References and neighboring frameworks

• Ho, Jain, and Abbeel, Denoising Diffusion Probabilistic Models, 2020.
• Song et al., Score-Based Generative Modeling through Stochastic Differential Equations, 2021.
• Kingma et al., Variational Diffusion Models, NeurIPS 2021.
• Lipman et al., Flow Matching for Generative Modeling, ICLR 2023.
• Liu, Gong, and Liu, Flow Straight and Fast, 2022.
• Tong et al., Improving and Generalizing Flow-Based Generative Models with Minibatch Optimal Transport, TMLR 2024.
• Albergo, Boffi, and Vanden-Eijnden, Stochastic Interpolants, JMLR 2025.
• Holderrieth et al., Generator Matching, ICLR 2025.
• De Bortoli et al., Diffusion Schrödinger Bridge, NeurIPS 2021.
• Xu et al., Poisson Flow Generative Models, 2022; and PFGM++, ICML 2023.
• Silveri, Conforti, and Durmus, Theoretical Guarantees in KL for Diffusion Flow Matching, 2024.
• Singh and Lagun, Latent Stochastic Interpolants, 2025.

This page is a research-oriented exposition of the BGM program and the current Markovization-gap paper. It deliberately distinguishes exact population claims, calibrated proxy claims, downstream evidence, and open extensions.