Deep Importance Sampling Using Tensor Trains with Application to a Priori and a Posteriori Rare Events
Deep Importance Sampling Using Tensor Trains with Application to a Priori and a Posteriori Rare Events
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DOI：
10.1137/23m1546981
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发表时间：
2022-09
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影响因子：
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通讯作者：
T. Cui;S. Dolgov;Robert Scheichl
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作者：
T. Cui;S. Dolgov;Robert Scheichl
我们提出了一种深度重要性抽样方法，适用于估计高维问题中的罕见事件概率。我们近似的最佳重要性分布在一般的重要性抽样问题作为一个参考分布下的组合保序变换，其中每个变换是由一个平方张量列车分解形成的推进。平方张量序列分解提供了一种可扩展的分析方法，用于通过密度近似构建保序高维变换。使用合成的地图移动沿着一系列的桥接密度简化的困难，直接近似集中的密度函数。为了计算未归一化概率分布的期望值，我们设计了一个比率估计器，该估计器使用单独的重要性分布来估计归一化常数，该重要性分布再次通过张量训练格式的变换组合来构建。与自规范化重要性抽样相比，这提供了更好的理论方差减少，从而为高效计算贝叶斯推理问题中的罕见事件概率打开了大门。微分方程约束的问题的数值实验表明，几乎没有增加的计算复杂性与事件概率为零，并允许计算迄今无法达到的估计罕见的事件概率为复杂的，高维的后验密度。
We propose a deep importance sampling method that is suitable for estimating rare event probabilities in high-dimensional problems. We approximate the optimal importance distribution in a general importance sampling problem as the pushforward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a squared tensor-train decomposition. The squared tensor-train decomposition provides a scalable ansatz for building order-preserving high-dimensional transformations via density approximations. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. To compute expectations over unnormalized probability distributions, we design a ratio estimator that estimates the normalizing constant using a separate importance distribution, again constructed via a composition of transformations in tensor-train format. This offers better theoretical variance reduction compared with self-normalized importance sampling, and thus opens the door to efficient computation of rare event probabilities in Bayesian inference problems. Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.