Optimally rotated coordinate systems for adaptive least-squares regression on sparse grids

Optimally rotated coordinate systems for adaptive least-squares regression on sparse grids
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稀疏网格上自适应最小二乘回归的最佳旋转坐标系

DOI：
10.1137/1.9781611975673.19
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
J. Oettershagen
J. Oettershagen
中科院分区：
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文献类型：
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作者：
B. Bohn;M. Griebel;J. Oettershagen

文献摘要

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对于具有大量数据点的低维数据集，标准核方法通常不再适用于回归。除了简单的线性模型或涉及的启发式深度学习模型之外，更大（内核）模型类的基于网格的离散化会导致算法，这些算法自然会在数据点的数量上线性扩展。对于中维或高维回归任务，这些基于网格的离散化遭受维数灾难。在这里，稀疏网格方法已被证明在很大程度上规避了这个问题。在这种情况下，空间和维度自适应稀疏网格，它可以检测和利用一个给定的低有效维数的名义上高维数据，是特别成功的。然而，他们依赖于一个轴对齐的结构的解决方案和表现出的问题，主要是倾斜和旋转coordination.In本文中，我们提出了一种预处理方法，这些自适应稀疏网格算法，确定一个优化的，问题相关的坐标系，从而降低了有效的维数给定的数据集的方差分析意义。我们提供了合成数据以及现实世界的数据的数值例子，以显示如何从我们的预处理方法的自适应稀疏网格最小二乘算法的好处。
For low-dimensional data sets with a large amount of data points, standard kernel methods are usually not feasible for regression anymore. Besides simple linear models or involved heuristic deep learning models, grid-based discretizations of larger (kernel) model classes lead to algorithms, which naturally scale linearly in the amount of data points. For moderate-dimensional or high-dimensional regression tasks, these grid-based discretizations suffer from the curse of dimensionality. Here, sparse grid methods have proven to circumvent this problem to a large extent. In this context, space- and dimension-adaptive sparse grids, which can detect and exploit a given low effective dimensionality of nominally high-dimensional data, are particularly successful. They nevertheless rely on an axis-aligned structure of the solution and exhibit issues for data with predominantly skewed and rotated coordinates.In this paper we propose a preprocessing approach for these adaptive sparse grid algorithms that determines an optimized, problem-dependent coordinate system and, thus, reduces the effective dimensionality of a given data set in the ANOVA sense. We provide numerical examples on synthetic data as well as real-world data to show how an adaptive sparse grid least squares algorithm benefits from our preprocessing method.