基于改进局部切空间排列的流形学习算法

局部切空间排列是一种广受关注的流形学习算法，其具备实现简单、全局最优等特点，但其难以有效处理稀疏采样或非均匀分布的高维观测数据。针对这一问题，该文提出一种改进的局部切空间排列算法。首先，提出一种基于L1范数的局部切空间估计方法，由于同时考虑了距离和结构因素，该方法得到的切空间较主成分分析方法更为准确。其次，在坐标排列步骤为了减小排列误差，设计了一种基于流形结构的加权坐标排列方案，并给出了具体的求解方法。基于人造数据和真实数据的实验表明，该算法能够有效地处理稀疏和非均匀分布的流形数据。

Manifold Learning Algorithm Based on Modified Local Tangent Space Alignment

The Local Tangent Space Alignment (LTSA) is one of the popular manifold learning algorithms since it is straightforward to implementation and global optimal. However, LTSA may fail when high-dimensional observation data are sparse or non-uniformly distributed. To address this issue, a modified LTSA algorithm is presented. At first, a new L1 norm based method is presented to estimate the local tangent space of the data manifold. By considering both distance and structure factors, the proposed method is more accurate than traditional Principal Component Analysis (PCA) method. To reduce the bias of coordinate alignment, a weighted scheme based on manifold structure is then designed, and the detailed solving method is also presented. Experimental results on both synthetic and real datasets demonstrate the effectiveness of the proposed method when dealing with sparse and non-uniformly manifold data.