论文标题
浓缩密度的低量张量重建，并应用于贝叶斯倒置
Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion
论文作者
论文摘要
运输图已成为一种流行的机制，可以通过优化的推动力使用样品传播来表达复杂的概率密度。除了其广泛的适用性和众所周知的成功之外，运输地图还遭受了几个缺点，例如优化过程引起的数值不准确性以及在关注数量时必须采用采样方案的事实，例如瞬间是要计算。本文为应对这些问题的概率密度函数（PDF）的准确功能近似提供了一种新颖的方法。通过将目标PDF通过不精确传输图解释为扰动的参考密度，以更容易访问的格式的后续功能表示可以有效地计算所需数量。我们在适当的坐标系中介绍了基于层的扰动参考密度的近似值，以将高维表示问题拆分为一组独立近似值，为此单独选择的正顺序基函数可用。这有效地激发了高维pDF的近似值H-和P-和P-和P-REFINEMENT（即``网格尺寸''和多项式程度的概念。为了规避维度的诅咒并能够无需采样访问一定量的兴趣，通过各种蒙特卡洛方法采用了张量列车格式的低级别重建。根据Hellinger距离和Kullback-Leibler Divergence得出了对开发方法的先验收敛分析。与蒙特卡洛和马尔可夫链蒙特卡洛方法相比，包括贝叶斯反问题和几个浓缩密度的应用阐明了（上）收敛。
Transport maps have become a popular mechanic to express complicated probability densities using sample propagation through an optimized push-forward. Beside their broad applicability and well-known success, transport maps suffer from several drawbacks such as numerical inaccuracies induced by the optimization process and the fact that sampling schemes have to be employed when quantities of interest, e.g. moments are to compute. This paper presents a novel method for the accurate functional approximation of probability density functions (PDF) that copes with those issues. By interpreting the pull-back result of a target PDF through an inexact transport map as a perturbed reference density, a subsequent functional representation in a more accessible format allows for efficient and more accurate computation of the desired quantities. We introduce a layer-based approximation of the perturbed reference density in an appropriate coordinate system to split the high-dimensional representation problem into a set of independent approximations for which separately chosen orthonormal basis functions are available. This effectively motivates the notion of h- and p-refinement (i.e. ``mesh size'' and polynomial degree) for the approximation of high-dimensional PDFs. To circumvent the curse of dimensionality and enable sampling-free access to certain quantities of interest, a low-rank reconstruction in the tensor train format is employed via the Variational Monte Carlo method. An a priori convergence analysis of the developed approach is derived in terms of Hellinger distance and the Kullback-Leibler divergence. Applications comprising Bayesian inverse problems and several degrees of concentrated densities illuminate the (superior) convergence in comparison to Monte Carlo and Markov-Chain Monte Carlo methods.
