Test-Time Anchoring for Discrete Diffusion Posterior Sampling

While continuous diffusion models have achieved remarkable success, discrete diffusion offers a unified framework for jointly modeling text and images. Beyond unification, discrete diffusion provides faster inference, finer control, and principled training-free guidance, making it well-suited for posterior sampling. Existing approaches to posterior sampling using discrete diffusion face severe challenges: derivative-free guidance yields sparse signals, continuous relaxations limit applicability, and split Gibbs samplers suffer from the curse of dimensionality. To overcome these limitations, we introduce Anchored Posterior Sampling (APS), built on two key innovations: quantized expectation for gradient-like guidance in discrete embedding space, and anchored remasking for adaptive decoding. APS achieves state-of-the-art performance among discrete diffusion samplers on both linear and nonlinear inverse problems across the standard image benchmarks. We demonstrate the generality of APS through training-free stylization and text-guided editing. We further apply APS to a large-scale diffusion language model, showing consistent improvement in question answering.

Big Picture Artificial intelligence models are increasingly moving toward a "discrete" approach—treating data as distinct building blocks, much like words in a sentence or individual puzzle pieces. This makes it easier for a single AI to understand and generate both text and images in a unified way. ### Problem Statement Often, we want to guide these pre-trained AI models to perform specific, complex tasks—such as fixing a corrupted image, editing a photo based on a text prompt, or making sure a language model gives a highly accurate answer. Doing this without spending massive amounts of time and money to retrain the AI from scratch is very difficult. Current methods for guiding discrete models are often clumsy, slow, or give the AI poor feedback, causing it to fail on larger, more complex tasks. ### Our Solution We introduce a new technique called Anchored Posterior Sampling (APS). Think of it as giving the AI a much smarter compass and a better eraser. APS works through two key innovations: It translates complex guidance into clear, step-by-step mathematical "hints" so the AI knows exactly how to improve its output. It gives the AI an adaptive way to systematically revisit and fix its own mistakes during the generation process. ### Why It Matters Our approach works incredibly well. APS sets a new state-of-the-art standard for guiding discrete AI models. We show that it can seamlessly restore images, apply artistic styles, and execute text-guided edits. Furthermore, APS isn't just for images—we applied it to a large-scale AI language model and proved it consistently improves the model's ability to accurately answer questions, all without any additional training.