An effective approach for sampling from unnormalized densities is based on the idea of gradually transporting samples from an easy prior to the complicated target distribution. Two popular methods are (1) Sequential Monte Carlo (SMC), where the transport is performed through successive annealed densities via prescribed Markov chains and resampling steps, and (2) recently developed diffusion-based sampling methods, where a learned dynamical transport is used. Despite the common goal, both approaches have different, often complementary, advantages and drawbacks. The resampling steps in SMC allow focusing on promising regions of the space, often leading to robust performance. While the algorithm enjoys asymptotic guarantees, the lack of flexible, learnable transitions can lead to slow convergence. On the other hand, diffusion-based samplers are learned and can potentially better adapt themselves to the target at hand, yet often suffer from training instabilities. In this work, we present a principled framework for combining SMC with diffusion-based samplers by viewing both methods in continuous time and considering measures on path space. This culminates in the new Sequential Controlled Langevin Diffusion (SCLD) sampling method, which is able to utilize the benefits of both methods and reaches improved performance on multiple benchmark problems, in many cases using only 10% of the training budget of previous diffusion-based samplers.

The single-cell RNA sequencing (scRNA-seq) technology enables researchers to study complex biological systems and diseases with high resolution. The central challenge is synthesizing enough scRNA-seq samples; insufficient samples can impede downstream analysis and reproducibility. While various methods have been attempted in past research, the resulting scRNA-seq samples were often of poor quality or limited in terms of useful specific cell subpopulations. To address these issues, we propose a novel method called Single-Cell Latent Diffusion (SCLD) based on the Diffusion Model. This method is capable of synthesizing large-scale, high-quality scRNA-seq samples, including both 'holistic' or targeted specific cellular subpopulations within a unified framework. A pre-guidance mechanism is designed for synthesizing specific cellular subpopulations, while a post-guidance mechanism aims to enhance the quality of scRNA-seq samples. The SCLD can synthesize large-scale and high-quality scRNA-seq samples for various downstream tasks. Our experimental results demonstrate state-of-the-art performance in cell classification and data distribution distances when evaluated on two scRNA-seq benchmarks. Additionally, visualization experiments show the SCLD's capability in synthesizing specific cellular subpopulations.