Amortized Posterior Sampling with Diffusion Prior Distillation

We propose a variational inference approach to sample from the posterior distribution for solving inverse problems. From a pre-trained diffusion model, our approach trains a conditional flow model to minimize the divergence between the proposal variational distribution and the posterior distribution implicitly defined through the diffusion model. Once trained, the flow model is capable of sampling from the posterior distribution with a single NFE, amortized with respect to the measurement. The proposed method paves a new path for distilling a diffusion prior for efficient posterior sampling. We show that our method is applicable to standard signals in Euclidean space, as well as signals on manifold.

Defining Neural Network Architecture through Polytope Structures of Dataset

Current theoretical and empirical research in neural networks suggests that complex datasets require large network architectures for thorough classification, yet the precise nature of this relationship remains unclear. This paper tackles this issue by defining upper and lower bounds for neural network widths, which are informed by the polytope structure of the dataset in question. We also delve into the application of these principles to simplicial complexes and specific manifold shapes, explaining how the requirement for network width varies in accordance with the geometric complexity of the dataset. Moreover, we develop an algorithm to investigate a converse situation where the polytope structure of a dataset can be inferred from its corresponding trained neural networks. Through our algorithm, it is established that popular datasets such as MNIST, Fashion-MNIST, and CIFAR10 can be efficiently encapsulated using no more than two polytopes with a small number of faces.