Jump Restore Light Transport

TL;DR

The paper tackles the inefficiency of traditional local-only MCMC for high-dimensional light transport by decoupling local exploration from global discovery. It introduces Jump Restore, a continuous-time MCMC framework that composes local Markov processes $Y^i$ with a global transfer mechanism $μ$ to yield a process invariant to the target distribution $π$. The method can embed and improve any existing MCMC-based light transport algorithm, delivering shorter runtimes, lower error, and reduced variance, while enabling parallel execution. Empirical results on rendering tasks and theoretical guarantees demonstrate superior mode coverage and fidelity, suggesting broad applicability to rendering and multimodal sampling challenges beyond graphics.

Abstract

Markov chain Monte Carlo (MCMC) algorithms are indispensable when sampling from a complex, high-dimensional distribution by a conventional method is intractable. Even though MCMC is a powerful tool, it is also hard to control and tune in practice. Simultaneously achieving both rapid local exploration of the state space and efficient global discovery of the target distribution is a challenging task. In this work, we introduce a novel continuous-time MCMC formulation to the computer science community. Generalizing existing work from the statistics community, we propose a novel framework for adjusting an arbitrary family of Markov processes - used for local exploration of the state space only - to an overall process which is invariant with respect to a target~distribution. To demonstrate the potential of our framework, we focus on a simple, but yet insightful, application in light transport simulation. As a by-product, we introduce continuous-time MCMC sampling to the computer graphics community. We show how any existing MCMC-based light transport algorithm can be seamlessly integrated into our framework. We prove empirically and theoretically that the integrated version is superior to the ordinary algorithm. In fact, our approach will convert any existing algorithm into a highly parallelizable variant with shorter running time, smaller error and less variance.

Figures (9)

• Figure 1: Histogram of Metropolis without and with a mixture proposal. Target: 1-dimensional Gaussian mixture with three modes. (a--c) Traditional Metropolis with local Gaussian proposals $\TextOrMath{$ζ$\xspace}{\zeta}(\TextOrMath{$x$\xspace}{x},\;\cdot\;)=\mathcal{N}_{\varsigma^2}(\TextOrMath{$x$\xspace}{x},\;\cdot\;)$ ($\TextOrMath{$ℓ$\xspace}{\ell}=0$), initialized near the left, center and right mode, respectively — each chain gets trapped in the mode nearest to its initial state. (d) Mixture proposal $\TextOrMath{$Q$\xspace}{Q}(\TextOrMath{$x$\xspace}{x},\;\cdot\;)=\TextOrMath{$ℓ$\xspace}{\ell}\TextOrMath{$μ$\xspace}{\mu}+(1-\TextOrMath{$ℓ$\xspace}{\ell})\TextOrMath{$ζ$\xspace}{\zeta}(\TextOrMath{$x$\xspace}{x},\;\cdot\;)$ with $\TextOrMath{$μ$\xspace}{\mu}$ uniform on the support and $\TextOrMath{$ℓ$\xspace}{\ell}=0.3$ enables large jumps across modes; the histogram now explores all three modes.
• Figure 2: We introduce a novel continuous-time MCMC framework that adjusts an arbitrary family of Markov processes $\TextOrMath{$Y$\xspace}{Y}^\TextOrMath{$i$\xspace}{i}$ — used solely for local exploration — to an overall process which is invariant with respect to a target distribution. Global discovery is achieved through a transfer mechanism $\TextOrMath{$μ$\xspace}{\mu}$. This mechanism interrupts local exploration immediately before an exponential clock $\TextOrMath{$τ$\xspace}{\tau}_\TextOrMath{$i$\xspace}{i}$ --- whose rate is inversely proportional to the target density --- expires, and then transfers the local exploration to a different region of the state space.
• Figure 3: Jump Restore algorithm with local dynamics, uniform global dynamics and the multimodal target distribution from \ref{['fig:global-discovery']}
• Figure 4: Equal rendering time comparison (20s) of (left), Restore (middle), and (right) for the Veach, ajar scene provided by lmc.
• Figure 5: $L^2$-error and empirical variance over rendering time in seconds for the Veach, ajar and Torus scene depicted in \ref{['fig:veach']} and \ref{['fig:torus']}, respectively.

Theorems & Definitions (4)

• Definition 4.1
• Definition 6.1
• Definition 7.1
• Definition 7.2