Weaving order from uncertainty: Design, Analysis, and Applications of Transport-based Generative Models

Abstract

Generative machine learning algorithms are pivotal for advancing artificial intelligence and for gaining insights into biological neural systems. In this thesis, we present a comprehensive study that integrates theoretical analysis, efficient algorithm development, and biological applications in generative modeling. We first establish a general theoretical framework for minimum divergence transport estimators, a prominent class of generative models. We derive a priori error bounds that quantify the generalization performance of these models in terms of model and sample complexity. Building upon this theory, we introduce a flow-based transport algorithm for generative modeling that utilizes kernel methods to minimize Maximum Mean Discrepancy (MMD). Our method offers an efficient alternative to existing neural network approaches, achieving comparable performance with fewer parameters and reduced training time. We apply the theoretical results from the first part to derive generalization bounds for this algorithm. Finally, we explore the intersection of generative modeling and neuroscience by developing a generative model for receptive fields in sensory neuronal systems using Gaussian processes. This model elucidates how sensory neurons transform inputs to create robust representations. Our biological insights inspire an initialization strategy that improves the efficiency of neural network training.