Prediction Targets in Diffusion Models: Geometry, Capacity, and Shortcuts

Although $x$-, $\epsilon$-, and $v$-prediction yield the same score function at optimum, their prediction targets can have markedly different geometric structures. Empirical studies have reported substantial performance gaps between these parameterizations in both pixel and representation spaces. We study this discrepancy through the interaction between prediction-target geometry and limited output rank. For data with low-dimensional structure, the clean predictor follows the underlying signal structure, whereas noise and velocity predictors additionally carry ambient components inherited from the noisy input. These additional components can create ambient approximation floors for $\epsilon$- and $v$-prediction, even when the same output rank is sufficient for $x$-prediction. This geometric characterization also suggests a simple time-dependent shortcut $c(t)x_t$ that directly carries the ambient component, leaving the core network to represent a scaled conditional clean signal. With the appropriate coefficient, the shortcut removes the resulting ambient approximation floor. Controlled experiments validate the predicted approximation scaling and evaluate the learned shortcut on nonlinear synthetic data. On ImageNet $256\times 256$ with JiT-B/16, the shortcut reduces $v$-prediction FID from $96.53$ to $7.83$, compared with $8.62$ for $x$-prediction.