Tensor-Train格式中近似高斯密度的秩界
Rank Bounds for Approximating Gaussian Densities in the Tensor-Train Format
摘要：
低秩张量近似已经显示出在高维度中用于不确定性量化的巨大潜力，例如，以构建可用于加速大规模推理问题的代理模型（Eigel等人，逆问题34，2018; Dolgov等人，统计与计算30，2020）。这种方法的可行性和效率主要取决于表示或近似基本分布所需的秩。在本文中，先验秩界近似的函数张量列车表示的情况下，高斯模型的发展。结果表明，在适当的条件下的精度矩阵，高斯密度可以近似到高精度，而不会遭受指数增长性的复杂性随着维数的增加。这些结果提供了一个严格的理由的适用性和低秩张量方法的局限性，在一个简单但重要的模型的情况下。数值实验证实，当精度矩阵的参数和近似的精度变化时，秩界捕获秩结构的定性行为。最后，在贝叶斯过滤问题的背景下，证明了理论结果的实际相关性。
英文摘要：
Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up large-scale inference problems (Eigel et al., Inverse Problems 34, 2018; Dolgov et al., Statistics & Computing 30, 2020). The feasibility and efficiency of such approaches depends critically on the rank that is necessary to represent or approximate the underlying distribution. In this paper, a-priori rank bounds for approximations in the functional tensor-train representation for the case of Gaussian models are developed. It is shown that under suitable conditions on the precision matrix, the Gaussian density can be approximated to high accuracy without suffering from an exponential growth of complexity as the dimension increases. These results provide a rigorous justification of the suitability and the limitations of low-rank tensor methods in a simple but important model case. Numerical experiments confirm that the rank bounds capture the qualitative behavior of the rank structure when varying the parameters of the precision matrix and the accuracy of the approximation. Finally, the practical relevance of the theoretical results is demonstrated in the context of a Bayesian filtering problem.
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