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.

翻译：本文提出一种适用于高维问题中罕遇事件概率估计的深度重要性采样方法。我们将一般重要性采样问题中的最优重要性分布近似为参考分布在若干保序变换复合作用下的前推分布，其中每个变换通过平方张量列分解构造。平方张量列分解为通过密度近似构建高维保序变换提供了可扩展的拟合法。采用沿桥接密度序列移动的复合映射策略，缓解了直接逼近集中密度函数的困难。为计算非归一化概率分布的期望，我们设计了一种比率估计器，通过独立的重要性分布（同样采用张量列格式的复合变换构造）估计归一化常数。该方法相比自归一化重要性采样具有更优的理论方差缩减效果，从而为贝叶斯推断问题中罕遇事件概率的高效计算开辟了途径。在微分方程约束问题上的数值实验表明，当事件概率趋近于零时，计算复杂度几乎不增加，并首次实现了对复杂高维后验密度中罕遇事件概率的估计。