Many recent works utilize denoising score matching to optimize the conditional input of diffusion models. In this workshop paper, we demonstrate that such optimization breaks the equivalence between denoising score matching and exact score matching. Furthermore, we show that this bias leads to higher score norm. Additionally, we observe a similar bias when optimizing the data distribution using a pre-trained diffusion model. Finally, we discuss the wide range of works across different domains that are affected by this bias, including MAR for auto-regressive generation, PerCo for image compression, and DreamFusion for text to 3D generation.

Score matching enables the estimation of the gradient of a data distribution, a key component in denoising diffusion models used to recover clean data from corrupted inputs. In prior work, a heuristic weighting function has been used for the denoising score matching loss without formal justification. In this work, we demonstrate that heteroskedasticity is an inherent property of the denoising score matching objective. This insight leads to a principled derivation of optimal weighting functions for generalized, arbitrary-order denoising score matching losses, without requiring assumptions about the noise distribution. Among these, the first-order formulation is especially relevant to diffusion models. We show that the widely used heuristical weighting function arises as a first-order Taylor approximation to the trace of the expected optimal weighting. We further provide theoretical and empirical comparisons, revealing that the heuristical weighting, despite its simplicity, can achieve lower variance than the optimal weighting with respect to parameter gradients, which can facilitate more stable and efficient training.

The density estimation is one of the core problems in statistics. Despite this, existing techniques like maximum likelihood estimation are computationally inefficient due to the intractability of the normalizing constant. For this reason an interest to score matching has increased being independent on the normalizing constant. However, such estimator is consistent only for distributions with the full space support. One of the approaches to make it consistent is to add noise to the input data which is called Denoising Score Matching. In this work we derive analytical expression for the Denoising Score matching using the Kernel Exponential Family as a model distribution. The usage of the kernel exponential family is motivated by the richness of this class of densities. To tackle the computational complexity we use Random Fourier Features based approximation of the kernel function. The analytical expression allows to drop additional regularization terms based on the higher-order derivatives as they are already implicitly included. Moreover, the obtained expression explicitly depends on the noise variance, so the validation loss can be straightforwardly used to tune the noise level. Along with benchmark experiments, the model was tested on various synthetic distributions to study the behaviour of the model in different cases. The empirical study shows comparable quality to the competing approaches, while the proposed method being computationally faster. The latter one enables scaling up to complex high-dimensional data.