MCMC: Bridging Rendering, Optimization and Generative AI

TL;DR

This work frames Markov Chain Monte Carlo (MCMC) as a unifying framework across rendering, optimization, and generative AI, emphasizing diffusion models and gradient-based methods for sampling from high-dimensional distributions. It surveys the theoretical foundations (SDEs, Langevin and Hamiltonian dynamics, and Monte Carlo integration) and surveys practical MCMC algorithms (MH, LMC, ALMC, HMC) with their tradeoffs. It then details MCMC applications in physically based rendering (path-space, MIS, MLT, and primary-sample approaches), optimization (Bayesian inference, SGD/SGLD, MAP), and generative modeling (VAEs, HVAEs, diffusion/ELBO connections, EBMs, and score-based methods). The synthesis highlights how MCMC can bridge diffusion, energy-based modeling, and SGD-based optimization, while noting challenges like reversibility constraints and the need for globally exploring proposals. The course articulates a path toward integrating probabilistic sampling with modern rendering and diffusion-based generative modeling, offering practical demonstrations and a software resource for researchers and practitioners.

Abstract

Generative artificial intelligence (AI) has made unprecedented advances in vision language models over the past two years. During the generative process, new samples (images) are generated from an unknown high-dimensional distribution. Markov Chain Monte Carlo (MCMC) methods are particularly effective in drawing samples from such complex, high-dimensional distributions. This makes MCMC methods an integral component for models like EBMs, ensuring accurate sample generation. Gradient-based optimization is at the core of modern generative models. The update step during the optimization forms a Markov chain where the new update depends only on the current state. This allows exploration of the parameter space in a memoryless manner, thus combining the benefits of gradient-based optimization and MCMC sampling. MCMC methods have shown an equally important role in physically based rendering where complex light paths are otherwise quite challenging to sample from simple importance sampling techniques. A lot of research is dedicated towards bringing physical realism to samples (images) generated from diffusion-based generative models in a data-driven manner, however, a unified framework connecting these techniques is still missing. In this course, we take the first steps toward understanding each of these components and exploring how MCMC could potentially serve as a bridge, linking these closely related areas of research. Our course aims to provide necessary theoretical and practical tools to guide students, researchers and practitioners towards the common goal of generative physically based rendering. All Jupyter notebooks with demonstrations associated to this tutorial can be found on the project webpage: https://sinbag.github.io/mcmc/

Figures (14)

• Figure 1: This course provides an overview of MCMC methods, which are powerful tools for sampling from complex distributions. An example of such a distribution is shown in the middle row (Part II). These complex distributions are common in physically based rendering and generative modeling. Additionally, we will review the impact of MCMC methods on gradient-based optimization techniques, which typically aim to find good local (or global) minima in the optimization landscape (inset on the middle-right).
• Figure 2: To create a Markov chain, we start with a random sample (a) that is generated with a given proposal (shown in orange). (b) This sample becomes the current state. The next sample (new state) is then generated from the current state following the proposal. (c) As the chain continues to grow, we obtain a sequence of samples that represents a Markov chain. (d) When run long enough, this Markov chain can generate samples that matches the target distribution.
• Figure 3: Brownian motion for different target densities. Simulating Brownian motion is the same as performing a random walk (a). However, we can restrict the motion along certain trajectories (b-e). The red dot represents the starting point, and the blue dashed line (b,d) or the blue region (c) represents the targetted region/trajectory. The motion is simulated to partially fill the space to demonstrate the evolution of the Markov chain.
• Figure 4: Access to the gradients (vector field) of the target distribution (a) is a natural information we can have for exploration. Most Markov transitions are diffusive in nature, i.e., the they can spend too much time near the initial point. In order to make large jumps away from the initial point, and into new, unexplored regions of the typical region, we need to exploit information about the geometry. Hamiltonian dynamics is the unique procedure for automatically generating this coherent exploration for sufficiently well-behaved target distributions. It not only allows efficient movement in the neighborhood of a mode (b) but also towards the mode that needs to be explored (c). The figure is inspired from betancourt2018conceptual.
• Figure 5: Illustration of a simple light path with four vertices and its corresponding weighting function.