The ε-Bridge: Dimension-Dependent Bridge-Noise Selection for Stochastic Interpolants

Choosing between deterministic Flow Matching (FM) and stochastic bridge-based methods is a basic design decision in continuous-time generative modeling, and the two perform differently across tasks. Stochastic interpolants place both in a common family, yet there is little quantitative guidance on when, and how much, bridge noise helps. We study this question through the $\epsilon$-bridge, a one-parameter Brownian-bridge stochastic-interpolant family that recovers FM at $\epsilon =0$. We show that the interpolant, conditional velocity, and fixed-predictor objective converge to their FM counterparts as $\epsilon \to 0$ (Theorem A); that the raw bridge correction energy grows linearly in the ambient dimension (Proposition B); and that, under explicit dimension-uniform posterior-moment and contraction conditions, the marginal-velocity regularity bound contains no trace factor (Theorem C). The asymmetry between these two scalings motivates a two-term error model whose minimizer yields a dimension-dependent noise-selection rule (Model D). Under a fixed training and evaluation protocol on Gaussian mixtures, the selected noise decreases with dimension and reaches the FM endpoint at a crossover that is reproduced on an independent target family, and it moves toward zero as the inference budget grows. PCA-CIFAR, pixel-space image, tabular, molecular, robotics, and point-cloud experiments show the same ordering along dimension, but no single calibrated boundary transfers between families. The crossover instead tracks the fit quality of the FM baseline: bridge noise helps essentially only where that baseline is well fit, and changing model capacity at fixed dimension moves the boundary accordingly. Dimension is therefore a search coordinate, and the FM baseline is an operational diagnostic of whether the search is worth running.