Diffusion Probabilistic Models (DPMs) have achieved strong generative performance, yet their inductive biases remain largely implicit. In this work, we aim to build inductive biases into the training and sampling of diffusion models to better accommodate the target distribution of the data to model. We introduce an anisotropic noise operator that shapes these biases by replacing the isotropic forward covariance with a structured, frequency-diagonal covariance. This operator unifies band-pass masks and power-law weightings, allowing us to emphasize or suppress designated frequency bands, while keeping the forward process Gaussian. We refer to this as spectrally anisotropic Gaussian diffusion (SAGD). In this work, we derive the score relation for anisotropic covariances and show that, under full support, the learned score converges to the true data score as $t\!\to\!0$, while anisotropy reshapes the probability-flow path from noise to data. Empirically, we show the induced anisotropy outperforms standard diffusion across several vision datasets, and enables selective omission: learning while ignoring known corruptions confined to specific bands. Together, these results demonstrate that carefully designed anisotropic forward noise provides a simple, yet principled, handle to tailor inductive bias in DPMs.

翻译：扩散概率模型（DPMs）已展现出强大的生成性能，但其归纳偏置在很大程度上仍隐含于模型之中。本研究旨在将归纳偏置显式地构建于扩散模型的训练与采样过程中，以更好地适配目标数据分布。我们引入了一种各向异性噪声算子，通过将各向同性的前向协方差替换为结构化的频率对角协方差来塑造这些偏置。该算子统一了带通掩码与幂律加权机制，使我们能够在保持前向过程为高斯分布的前提下，强调或抑制指定的频带。我们将此方法称为谱各向异性高斯扩散（SAGD）。本文推导了各向异性协方差下的分数关系，并证明在完全支撑的条件下，当$t\!\to\!0$时，所学分数收敛于真实数据分数，而各向异性则重塑了从噪声到数据的概率流路径。实验表明，在多个视觉数据集上，所引入的各向异性优于标准扩散模型，并能实现选择性忽略：在学习过程中排除局限于特定频带的已知伪影。综上，这些结果证明，精心设计的各向异性前向噪声为定制DPMs的归纳偏置提供了一个简洁而原理清晰的控制机制。