| dc.description.abstract | Recent advances in score-based (diffusion) generative models have achieved state-of-the-art sample quality across standard benchmarks. Building on the remarkable property of these models in estimating scores, this thesis presents three core contributions: 1) new objectives to reduce score estimation error, 2) a novel Bayesian-inspired optimization framework for solving inverse problems, and 3) a fast one-step generative modeling framework that is based on a novel amortized score estimation framework. In the first part of this thesis, we introduce two new score estimation objectives with applications to both implicit and diffusion-based generative models. To improve spectralbased non-parametric estimators, we propose a theoretically optimal parametric framework that learns the score by projecting it onto its top-L principal directions. Additionally, inspired by matrix-valued kernel methods, we present a second approach that lifts the score into the space of outer products, and minimizes the distance between the estimated and true scores in this higher-order space. In the second part, we shift focus from score estimation to leveraging diffusion models as data-driven priors for solving inverse problems. Centering our development around the problem of source separation, we introduce a novel algorithm inspired by maximum a posteriori estimation. This approach combines multiple levels of Gaussian smoothing with an α-posterior, enabling effective signal separation using only independent priors for the sources. We demonstrate the effectiveness of this method through its application to interference mitigation in digital communication signals. Finally, we outline how this framework can be naturally extended to tackle a broader class of inverse problems. In the final part, we return to the fundamental challenge of efficient sampling, which is critical for enabling practical data-driven engineering systems. We propose a novel generative modeling framework that enables training a one-step neural sampler from scratch. At the core of this method is a new objective based on multi-divergence minimization, guided by a novel approach for score estimation of mixture distributions. Our framework is simple to implement, stable during training, unifies several existing approaches, and achieves state-of-the-art performance in image generation tasks. Furthermore, we discuss how this framework can be naturally extended to multi-step neural sampling and adapted for fast posterior sampling—an essential component in simulation-based inverse problem solvers. | |
