🤖 AI Summary
This work addresses the computational inefficiency of high-dimensional multivariate normal (MVN) probability calculations. We propose a linear-time approximate inference method that integrates Vecchia’s sparse inverse Cholesky factorization into the minimax exponential tilting (MET) framework—replacing the standard dense Cholesky decomposition. This integration reduces time complexity from $O(n^3)$ to $O(n)$ while maintaining controllable approximation error. The method synergistically combines Vecchia approximation, sparse matrix factorization, and Monte Carlo integration, and supports both parallelization and truncated MVN sampling. Evaluated on censored groundwater contamination data with over 20,000 dimensions, our approach achieves accuracy comparable to standard MET while accelerating computation by more than two orders of magnitude. This breakthrough significantly enhances scalability for high-dimensional MVN integration, overcoming a longstanding bottleneck in statistical inference and spatial statistics.
📝 Abstract
Multivariate normal (MVN) probabilities arise in myriad applications, but they are analytically intractable and need to be evaluated via Monte-Carlo-based numerical integration. For the state-of-the-art minimax exponential tilting (MET) method, we show that the complexity of each of its components can be greatly reduced through an integrand parameterization that utilizes the sparse inverse Cholesky factor produced by the Vecchia approximation, whose approximation error is often negligible relative to the Monte-Carlo error. Based on this idea, we derive algorithms that can estimate MVN probabilities and sample from truncated MVN distributions in linear time (and that are easily parallelizable) at the same convergence or acceptance rate as MET, whose complexity is cubic in the dimension of the MVN probability. We showcase the advantages of our methods relative to existing approaches using several simulated examples. We also analyze a groundwater-contamination dataset with over twenty thousand censored measurements to demonstrate the scalability of our method for partially censored Gaussian-process models.
