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：在二次流形上实现降维，通过二次修正项增强线性逼近。先前的工作依赖于训练数据前几个主成分投影得到的线性逼近；然而，仅由前几个主成分张成的子空间中的线性逼近可能遗漏对二次修正项有效性至关重要的信息。本文提出一种贪心方法，从前序及后续主成分中构造子空间，使得相应的线性逼近能通过二次项实现最高效的修正。贪心构造流形的性质允许应用线性代数重表述，从而使该方法可扩展至数百万维度的数据点。数值实验表明，与仅基于前几个主成分的流形相比，贪心构造的二次流形可实现数量级更高的精度。