Rank Bounds for Approximating Gaussian Densities in the Tensor-Train Format
Rank Bounds for Approximating Gaussian Densities in the Tensor-Train Format
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DOI：
10.1137/20m1314653
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发表时间：
2020-01
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通讯作者：
Paul B. Rohrbach;S. Dolgov;L. Grasedyck;Robert Scheichl
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文献类型：
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作者：
Paul B. Rohrbach;S. Dolgov;L. Grasedyck;Robert Scheichl
例如，低秩张量近似已被成功地用于构建代理模型，该模型可用于加速高维的大规模推理问题。这种方法的成功与否主要取决于表示或近似潜在分布所必需的等级。在本文中，我们开发了高斯（正态分布）模型的泛函张量-训练表示近似的先验秩界。我们证明了在合适的精度矩阵条件下，我们可以高精度地表示高斯密度，而不会受到复杂度随维数增加而呈指数增长的影响。我们的结果在一个简单但重要的模型案例中证明了低秩张量方法的适用性和局限性。数值实验证实，在改变精度矩阵参数和近似精度的情况下，秩界捕捉到了秩结构的定性行为。
Low rank tensor approximations have been employed successfully, for example, to build surrogate models that can be used to speed up large-scale inference problems in high dimensions. The success of this depends critically on the rank that is necessary to represent or approximate the underlying distribution. In this paper, we develop a-priori rank bounds for approximations in the functional Tensor-Train representation for the case of a Gaussian (normally distributed) model. We show that under suitable conditions on the precision matrix, we can represent the Gaussian density to high accuracy without suffering from an exponential growth of complexity as the dimension increases. Our results provide evidence of the suitability and 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.