利用Rosenblatt输运的平方逆张量序列的深层合成
Deep composition of tensor-trains using squared inverse Rosenblatt transports
摘要：
刻画高维随机变量是随机计算的基本挑战之一。最近激增的运输地图提供了一个数学基础和新的见解，通过耦合棘手的随机变量与易处理的参考随机变量来应对这一挑战。本文将Dolgov等人（Stat Comput 30：603- 625，2020）最近提出的逆Rosenblatt输运的泛函张量列近似推广到一类高维非负函数，如非归一化概率密度函数。首先，我们扩展逆Rosenblatt变换，使运输一般的参考措施以外的统一措施。我们开发了一个有效的程序来计算这种运输从平方张量列车分解，保持单调性。更重要的是，我们将所提出的保序功能张量火车运输到一个嵌套的变量转换框架的启发，深层神经网络的分层结构。由此产生的深逆Rosenblatt输运显着扩展张量近似和输运映射到具有复杂非线性相互作用和集中密度函数的随机变量的能力。我们证明了所提出的方法在统计学习和不确定性量化中的一系列应用的效率，包括动力系统的参数估计和偏微分方程约束的反问题。
英文摘要：
Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. The recent surge of transport maps offers a mathematical foundation and new insights for tackling this challenge by coupling intractable random variables with tractable reference random variables. This paper generalises the functional tensor-train approximation of the inverse Rosenblatt transport recently developed by Dolgov et al. (Stat Comput 30:603--625, 2020) to a wide class of high-dimensional non-negative functions, such as unnormalised probability density functions. First, we extend the inverse Rosenblatt transform to enable the transport to general reference measures other than the uniform measure. We develop an efficient procedure to compute this transport from a squared tensor-train decomposition which preserves the monotonicity. More crucially, we integrate the proposed order-preserving functional tensor-train transport into a nested variable transformation framework inspired by the layered structure of deep neural networks. The resulting deep inverse Rosenblatt transport significantly expands the capability of tensor approximations and transport maps to random variables with complicated nonlinear interactions and concentrated density functions. We demonstrate the efficiency of the proposed approach on a range of applications in statistical learning and uncertainty quantification, including parameter estimation for dynamical systems and inverse problems constrained by partial differential equations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
