Lower-dimensional principal manifold learning in higher-dimensional data spaces by sparse grid methods

在许多科学领域中，可以观察到高维问题只有低有效维。这可以用来规避数值近似的维数诅咒。然而，主要的问题是如何找到合适的坐标系和最佳的关联有效维数。这是特别困难的，如果一个允许非线性的内在坐标，然而，往往允许大大优于线性方法的PCA降维。在这个项目中，我们使用正则化主流形的方法来计算这样的坐标。由于剩余的维度可能超过三个，我们使用稀疏网格来表示流形的坐标函数。通过这种方法，进一步降低了数值复杂性，并且可以有效地处理多达20个内在维的问题。利用维数敏感误差指标，采用维数自适应稀疏网格法自动导出有效维数和固有坐标。

In many areas of science, it can be observed that problems of high dimensions possess only low effective dimensionality. This may be exploited to circumvent the curse of dimensionality of a numerical approximation. The main problem, however, is to find a proper coordinate system and the best associated effective dimension. This is especially difficult if one allows nonlinear intrinsic coordinates which however often allow for a substantially better dimension reduction than a linear approach by the PCA. In this project, we use the approach of regularized principal manifolds to compute such coordinates. Since the remaining dimensions may nevertheless be more than three, we employ sparse grids for the representation of the coordinate functions of the manifold. This way, the numerical complexity is further reduced and problems with up to 20 intrinsic dimensions may be handled efficiently. We will derive the effective dimension and the intrinsic coordinates automatically by a dimension-adaptive sparse grid method using dimension sensitive error indicators.