Academic

Statistical inference from high-dimensional data with low-dimensional structures has recently attracted a lot of attention. In machine learning, deep generative modelling approaches implicitly estimate distributions of complex objects by creating new samples from the underlying distribution, and have achieved great success in generating synthetic realistic-looking images and texts. A key step in these approaches is the extraction of latent features or representations (encoding) that can be used for accurately reconstructing the original data (decoding). In other words, low-dimensional manifold structure is implicitly assumed and utilized in the distribution modelling and estimation. To understand the benefit of low-dimensional manifold structure in generative modelling, we build a general minimax framework for distribution estimation on unknown submanifold under adversarial losses, with suitable smoothness assumptions on the target distribution and the manifold. Through the perspective of minimax rates, we examine the statistical capabilities of the popular score-based generative models for adapting to the data manifold structure.