Tail-Preserving Diffusion via Geometric Regularization

Class-conditional diffusion models trained on long-tailed data often generate minority-class samples that drift toward head-class semantics. We study this failure through the distinction between conditional geometry preservation and marginal coverage control. Our analysis shows that positive class-only reweighting preserves the population-optimal conditional denoising target, although it can alter shared-parameter optimization. Sampling-prior adjustment changes the marginal mixture while leaving each conditional distribution fixed. Under a local manifold model, we further bound deterministic reverse-trajectory departure in terms of initial distance and normal velocity, motivating geometric control of semantic drift. Guided by this analysis, we propose Geometry-aware Tail-preserving Score Regularization (GTSR). GTSR retains the standard denoising objective and adds a tail-specific regularizer on denoised estimates in a frozen semantic representation. A real-sample memory bank and rank-adaptive kNN-PCA define a local class reference, and the regularizer penalizes the normal-displacement energy fraction as a practical geometric proxy. Low-noise activation selects estimates that support reliable comparison with this reference. Evaluation on CIFAR10-LT, CIFAR100-LT, and Tiny-ImageNet-LT shows improved tail conditional accuracy, conditional fidelity, and semantic isolation, with consistent gains under DDPM and EDM parameterizations. These findings support geometry-preserving training complemented by sampling-time coverage control.