Spectral Diffusion Processes

Diffusion models have shown remarkable performance on diverse generation tasks. Recent work finds that imposing representation alignment on the hidden states of diffusion networks can both facilitate training convergence and enhance sampling quality, yet the mechanism driving this synergy remains insufficiently understood. In this paper, we investigate the connection between self-supervised spectral representation learning and diffusion generative models through a shared perspective on perturbation kernels. On the diffusion side, samples (e.g., images, videos) are produced by reversing a stochastic noise-injection process specified by Gaussian kernels; on the spectral representation side, spectral embeddings emerge from contrasting positive and negative relations induced by random perturbation kernels. Motivated by this, we propose a self-supervised spectral representation alignment method to facilitate diffusion model training. In addition, we clarify how joint spectral learning can benefit diffusion training from a geometric perspective. Furthermore, we find that the optimization of the spectral alignment objective is in an equivalent form of diffusion score distillation in the representation space. Building on these findings, we integrate a spectral regularizer into diffusion training objectives to improve the performance of diffusion models on multiple datasets. Experiments across images and 3D point clouds show consistent gains in generation quality. Code is released at https://github.com/yuehaowang/spectral-reg-diffusion.

The performance of generative diffusion models is determined by the choice of the reference diffusion process connecting the empirical and prior distributions. Conventional approaches typically trade off simulation-free training against finite-time generation. We propose a framework for designing the reference process that achieves both simultaneously. The key idea is to prescribe tractable time-dependent conditional distributions and then construct the reference process realizing them as its marginals. This framework reveals that score matching is not fundamental to diffusion-model training but instead emerges naturally through reversal of the reference process. We further show that conditional flow matching arises as the small-noise limit of the proposed framework.

We introduce Spectral Guidance, a framework for controlling diffusion models by leveraging the intrinsic geometry of the generative process. As data is progressively corrupted by noise, only a small number of features remain informative for control. We characterize them as the singular functions of a conditional expectation operator and show that they can be learned via a self-supervised objective. Once recovered, this basis enables the projection of arbitrary guidance signals, such as labels, CLIP embeddings, or masks, directly onto the sampling trajectory. This approach allows for stable, high-fidelity control without retraining or denoiser backpropagation during sampling. Empirically, we improve conditional accuracy on CIFAR-10 by 37 percentage points over the strongest training-free baseline while offering $4\times$ faster sampling. Moreover, the same representations that support label and CLIP guidance also enable spatial control, such as mask-based guidance, without auxiliary models. Finally, our framework reveals a phase transition in the generative process, pinpointing the optimal time window for effective guidance.