Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment
Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment
复制标题
DOI：
--
复制
发表时间：
2002-12
期刊：
影响因子：
--
通讯作者：
Zhenyue Zhang;Hongyuan Zha
中科院分区：
文献类型：
--
作者：
Zhenyue Zhang;Hongyuan Zha
无组织数据点的非线性流形学习是一个非常具有挑战性的无监督学习和数据可视化问题，有着广泛的应用。本文提出了一种新的流形学习和非线性降维算法。基于一组无组织的数据点采样与噪声从流形，我们表示的局部几何形状的流形使用切空间通过拟合仿射子空间在每个数据点的邻域学习。这些切空间被对齐，以通过邻域连接矩阵的部分特征分解给出数据点相对于底层流形的内部全局坐标。我们提出了一个仔细的误差分析，我们的算法，并表明重建误差是二阶精度。我们说明我们的算法使用曲线和曲面在2D/3D和高维欧氏空间，和64 × 64像素的人脸图像与各种姿势和照明条件。我们还解决了一些理论和算法问题，为进一步的研究和改进。
Nonlinear manifold learning from unorganized data points is a very challenging unsupervised learning and data visualization problem with a great variety of applications. In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction. Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood of each data point. Those tangent spaces are aligned to give the internal global coordinates of the data points with respect to the underlying manifold by way of a partial eigendecomposition of the neighborhood connection matrix. We present a careful error analysis of our algorithm and show that the reconstruction errors are of second-order accuracy. We illustrate our algorithm using curves and surfaces both in 2D/3D and higher dimensional Euclidean spaces, and 64-by-64 pixel face images with various pose and lighting conditions. We also address several theoretical and algorithmic issues for further research and improvements.