Scalable Marginalization of Correlated Latent Variables with Applications to Learning Particle Interaction Kernels
Scalable Marginalization of Correlated Latent Variables with Applications to Learning Particle Interaction Kernels
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DOI：
10.51387/22-nejsds13
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发表时间：
2022-03
期刊：
影响因子：
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通讯作者：
Mengyang Gu;Xubo Liu;X. Fang;Sui Tang
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文献类型：
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作者：
Mengyang Gu;Xubo Liu;X. Fang;Sui Tang
潜在变量或有害参数的边际化是贝叶斯推理和不确定性量化的一个基本方面。在这项工作中，我们专注于在相关数据建模中对潜在变量进行可扩展的边际化，例如时空或功能观测。我们首先介绍了用于相关数据建模的高斯过程(GP)，并强调了计算挑战，其中计算复杂性随着观测数量的增加而成倍增加。然后，我们回顾了状态空间模型和具有时间输入的Matérn协方差的GP之间的联系。作为一种可扩展的边际化技术，引入了卡尔曼滤波和Rauch-Toong-Striebel平滑器，用于在没有近似的情况下计算似然和预测GP。我们介绍了将可伸缩的边际化思想扩展到用于多变量相关输出和时空观测的协区域化的线性模型的最新努力。在这项工作的最后，我们介绍了一种新的边际化技术来估计相互作用核和预测粒子轨迹。计算过程在于对潜在变量的逆协方差矩阵进行稀疏表示，然后应用共轭梯度来提高大数据集的预测精度。在这项工作中取得的计算进展概述了在分子动力学模拟、细胞迁移和基于试剂的模型中的广泛应用。
Marginalization of latent variables or nuisance parameters is a fundamental aspect of Bayesian inference and uncertainty quantification. In this work, we focus on scalable marginalization of latent variables in modeling correlated data, such as spatio-temporal or functional observations. We first introduce Gaussian processes (GPs) for modeling correlated data and highlight the computational challenge, where the computational complexity increases cubically fast along with the number of observations. We then review the connection between the state space model and GPs with Matérn covariance for temporal inputs. The Kalman filter and Rauch-Tung-Striebel smoother were introduced as a scalable marginalization technique for computing the likelihood and making predictions of GPs without approximation. We introduce recent efforts on extending the scalable marginalization idea to the linear model of coregionalization for multivariate correlated output and spatio-temporal observations. In the final part of this work, we introduce a novel marginalization technique to estimate interaction kernels and forecast particle trajectories. The computational progress lies in the sparse representation of the inverse covariance matrix of the latent variables, then applying conjugate gradient for improving predictive accuracy with large data sets. The computational advances achieved in this work outline a wide range of applications in molecular dynamic simulation, cellular migration, and agent-based models.