Noise-Level KL Rates for Multi-Marginal Schrödinger Bridge Surrogates

It is now common to use 2D diffusion priors for problems where the underlying object is 3D (e.g., medical imaging). We would pick a noise schedule from the EDM literature, anneal a tilted prior, and then use the formulation as a 3D generative model. It works empirically. But we know very little about what it converges to and how fast. We study this question for a natural target: the multi-marginal Schrödinger bridge, the relative-entropy projection of a 3D prior onto the distribution whose rendered slices match the 2D marginals. But computing it is intractable. The annealed 2D-tilted surrogate is what one actually uses in applications. The main question is how close the latter object is to the former. Our main idea is that the discrepancy between these two laws — a comparison of two high-dimensional, fully coupled distributions — can be reduced to the size of a *single* scalar correction field. This field has a nice two-piece form: residual coupling across slices and tempering mismatch from the annealed discrete schedule. Using a Gaussian-smoothed filtration, Wasserstein screening, and conditional concentration, we obtain an explicit discrepancy bound in which the relevant terms separate into latent and posterior-residual contributions. When the smoothing scale is identified with the EDM noise level $\sigma(\tau)$, our bound yields $\mathrm{KL}(\Pi_\tau\|P^\star)=O\left(N\sigma(\tau)^2\right)$ and $\|\Pi_\tau-P^\star\|_{\mathrm{TV}}=O\left(\sqrt{N}\sigma(\tau)\right),$ if the latent scale remains bounded and the surrogate log-discrepancy is noise-aligned. So, the 2D-tilted construction converges to the Schrödinger-bridge lift at a rate governed by user-checkable quantities like the surrogate Lipschitz constants and the posterior concentration constant.