Dimensionality reduction on quadratic manifolds augments linear approximations with quadratic correction terms. Previous works rely on linear approximations given by projections onto the first few leading principal components of the training data; however, linear approximations in subspaces spanned by the leading principal components alone can miss information that are necessary for the quadratic correction terms to be efficient. In this work, we propose a greedy method that constructs subspaces from leading as well as later principal components so that the corresponding linear approximations can be corrected most efficiently with quadratic terms. Properties of the greedily constructed manifolds allow applying linear algebra reformulations so that the greedy method scales to data points with millions of dimensions. Numerical experiments demonstrate that an orders of magnitude higher accuracy is achieved with the greedily constructed quadratic manifolds compared to manifolds that are based on the leading principal components alone.

翻译：基于二次流形的降维方法通过引入二次修正项来增强线性近似的精度。已有研究通常依赖于将训练数据投影至前几个主要主成分所张成的子空间来获得线性近似；然而，仅基于主导主成分子空间的线性近似可能遗漏对二次修正项高效性至关重要的信息。本研究提出一种贪婪方法，该方法同时利用主导主成分与后续主成分构造子空间，使得对应的线性近似能够通过二次项实现最高效的修正。贪婪构造流形的特性允许采用线性代数重构，从而使该方法能够扩展到数百万维的数据点。数值实验表明，与仅基于主导主成分的流形相比，贪婪构造的二次流形可实现精度数量级的提升。