The rise of diffusion models in the field of machine learning is making significant strides in modeling complex data distributions and generating realistic samples from various domains, such as images, videos, audio, and 3D scenes. Nevertheless, full theoretical comprehension of generative diffusion models continues to be a challenging frontier requiring a more elaborate understanding, particularly in the context of high-dimensional data spaces. Addressing this notorious standard, called the curse of dimensionality, requires inventive tactics capable of simultaneously considering the large amount and dimensionality of the data.
Diffusion models operate in two stages: forward diffusion, where noise is progressively added till the data point becomes pure noise, and backward diffusion, whereby the image is denoised using an effective force field learned from techniques such as score matching and deep neural networks. Researchers are focusing on developing diffusion models that can establish the precise empirical score, typically achievable through extensive training of strongly overparameterized deep networks when the size of the dataset isn’t too large.
This research aims at characterizing the dynamics of diffusion models in the simultaneous limit of large dimensions and large datasets, resulting in a profound understanding of the dynamical regimes involved in the backward generative diffusion process. These regimes include pure Brownian motion, specialization towards main data classes, and ultimate collapse onto specific data points. Understanding these dynamics is paramount, particularly in assuring that the generative models avoid memorization of the training dataset, which, in turn, can lead to overfitting.
Interestingly, to bypass memorization mishap at finite times, the dataset size must be exponentially large to match the dimension. Alternatively, practical implementations lean towards regularization and the approximate learning of the score, departing from its exact form. These findings offer unique insights into the consequences of using the same empirical score framework by drawing upon characteristic cross-over times like speciation time and collapse time during the diffusion process.
Significantly, this research holds both theoretical and practical implications. The study underlines the functional relevance of the research and provides guidelines for future exploration beyond the exact empirical score framework by validating its academic findings through numerical experiments on real datasets like CIFAR-10, ImageNet, and LSUN. Such a thorough investigation and understanding of generative diffusion models offer a monumental stride forward in Machine Learning technologies.
