A mathematical study reports that a path which first adds noise, adjusts its midpoint and then runs diffusion backward can approximate an exact Schrödinger bridge as the temperature parameter becomes large. In the Euclidean Ornstein-Uhlenbeck setting, the symmetric relative entropy between the constructed path and the exact bridge remains bounded after multiplication by exp(T/2). On compact manifolds with a positive spectral gap, the corresponding quantity remains bounded after multiplication by exp(λ1 T).
The work is a theoretical probability note built around analytical theorem proving in the Euclidean Ornstein-Uhlenbeck and compact-manifold settings. Its analytic results assume endpoint measures with finite relative entropy relative to the reference measure and bounded, compactly supported densities.
A midpoint joins two kinds of motion
A dynamic Schrödinger bridge is a path measure chosen by minimizing relative entropy to a reference path law while fixing the endpoint marginals μ and ν. In ordinary terms, it compares whole trajectories but must match the prescribed distributions at the start and finish.
The proposed path measure, Q^T, first runs the reference process forward from μ, moves the halfway marginal through a quadratic-cost optimal transport map, and then runs the time-reversed reference process from ν. It is intended to resemble forward diffusion during noising and approximate time reversal during denoising.
For unequal endpoint measures, the denoising phase is explicitly only approximate rather than exact time reversal. The comparison with the exact bridge is made at the process level through symmetric relative entropy, a measure of discrepancy between the two path laws.
The spectrum sets the error scale
In the Euclidean case, the reference process is Ornstein-Uhlenbeck, and the theorem controls the symmetric relative entropy after exp(T/2) rescaling. The result is an asymptotic bound on the mismatch between the two path measures, rather than an empirical performance score.
On a compact manifold, the scale is exp(λ1 T), where λ1 is the positive spectral-gap term used in the analysis. After that rescaling, the symmetric relative entropy between Q^T and the exact bridge has a finite limsup. The result is conditional on the specified compact-manifold setting and its assumptions.
The analysis also gives an exact spectral next-order correction to the entropic cost after accounting for the endpoint relative entropies. The correction is determined by coefficients of the endpoint densities on the first eigenspace. In the compact-manifold case with equal endpoint measures, the rescaled symmetric relative entropy converges to the sum of squared coefficients on that eigenspace.
One construction, split at stationarity
On compact manifolds, Q^T can also be formed by gluing two half-horizon Schrödinger bridges along their shared stationary marginal. The resulting path law approximates the full-temperature bridge and has a finite limsup when its symmetric relative entropy is multiplied by exp(λ1 T).
An endpoint result with a narrower reach
An explicitly computed Gaussian endpoint example points to a faster endpoint-level scale. For endpoint measures πT and qT, the symmetric relative entropy remains bounded after exp(2T) rescaling. Because this calculation concerns endpoint distributions, it does not establish a faster rate for full path measures.
The assumptions set clear boundaries on the result. The main analytic statements are restricted to finite endpoint relative entropies and bounded, compactly supported densities in either the Euclidean Ornstein-Uhlenbeck or compact-manifold diffusion setting. The Gaussian illustration lies outside the theorem's compact-support condition, so its endpoint rate should not be read as a general path-level result.
The document is an arXiv preprint, version 1, dated 25 August 2026. Its evidence consists of analytical theorem proving for the stated settings, so the reported bounds describe asymptotic approximation in those models rather than a measured result from a generative-model evaluation.
Paper data and sources
Original title: Noising-Denoising by Large Temperature Schrödinger Bridges
Authors: Garrett Mulcahy
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text