Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports
Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports
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DOI：
10.1007/s10208-021-09537-5
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发表时间：
2020-07
影响因子：
3
通讯作者：
T. Cui;S. Dolgov
中科院分区：
文献类型：
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作者：
T. Cui;S. Dolgov
刻画难处理的高维随机变量是随机计算的基本挑战之一。最近运输地图的激增为通过将难处理的随机变量与易处理的参考随机变量相结合来应对这一挑战提供了数学基础和新的见解。本文推广了Dolgov等人最近提出的逆Rosenblatt输运的泛函张量-列近似。(Stat Comput30：603-625,2020)推广到一大类高维非负函数，如非归一化概率密度函数。首先，我们扩展了Rosenblatt逆变换，使之能够传输到一般参考测量而不是一致测量。我们从保持单调性的平方张量序列分解出发，发展了一种计算这种输运的有效方法。更重要的是，我们将所提出的保序泛函张量-训练传输集成到一个嵌套变量变换框架中，该框架的灵感来自于深层神经网络的分层结构。由此产生的深逆Rosenblatt输运极大地扩展了张量近似和输运映射到具有复杂的非线性相互作用和集中密度函数的随机变量的能力。我们在统计学习和不确定性量化的一系列应用中证明了该方法的有效性，包括动态系统的参数估计和偏微分方程组约束的反问题。
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.