基于局域主方向重构的适应性非线性维数约减

现有的主要非线性维数约减算法，如SIE和Isomap等，其邻域参数的设定是全局性的。仿真表明，对于局域流形结构差异较大的数据集，全局一致的邻域参数可能无法获得合理的嵌入结果。为此给出基于局域主方向重构的适应性邻域选择算法。算法首先为每个参考点选择一个邻域集，使各邻域集近似处于局域主线性子空间，并计算各邻域集的基向量集；再由基向量集对各邻域点的线性拟合误差判定该邻域点与主线性子空间的偏离程度，删除偏离较大的点。仿真表明，基于局域主方向重构的适应性邻域选择可有效处理局域流形结构差异较大的数据集；且相对于已有的适应性邻域选择算法，可以更好屏蔽靠近参考点的孤立噪声点及较大的空间曲率导致的虚假连通性。

Locally adaptive nonlinear dimensionality reduction

Popular nonlinear dimensionality reduction algorithms, such as SIE and Isomap suffer a difficulty in common: global neighborhood parameters often fail in tackling data sets with high variation in local manifold. To improve the availability of nonlinear dimensionality reduction algorithms in the field of machine learning, an adaptive neighbors selection scheme based on locally principal direction reconstruction was proposed.The method involves two main computation steps. First, it selects an appropriate neighborhood set for each data points such that all neighbors in a neighborhood set form a d-dimensionality linear subspace approximatively and computes locally principal directions for each neighborhood set respectively. Secondly, it fits each neighbor by means of locally principal directions of corresponding neighborhood set and deletes the neighbors whose fitting error exceed a predefined threshold. The simulation show that the method can deal with data set with high variation in local manifold effectively. Moreover, comparing with other adaptive neighbors selection strategy,this method can circumvent false connectivity introduced by noise or high local curvature.