Nonlinear manifold learning from unorganized data points is a verychallenging unsupervised learning and data visualization problem with a greatvariety of applications. In this paper we present a new algorithm for manifoldlearning and nonlinear dimension reduction. Based on a set of unorganized datapoints sampled with noise from the manifold, we represent the local geometry ofthe manifold using tangent spaces learned by fitting an affine subspace in aneighborhood of each data point. Those tangent spaces are aligned to give theinternal global coordinates of the data points with respect to the underlyingmanifold by way of a partial eigendecomposition of the neighborhood connectionmatrix. We present a careful error analysis of our algorithm and show that thereconstruction errors are of second-order accuracy. We illustrate our algorithmusing curves and surfaces both in 2D/3D and higher dimensional Euclidean spaces, and 64-by-64 pixel face imageswith various pose and lighting conditions. We also address several theoreticaland algorithmic issues for further research and improvements.
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机译：从无组织的数据点进行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，具有多种应用。在本文中，我们提出了一种用于流形学习和非线性降维的新算法。基于从流形噪声中采样的一组无序数据点，我们使用切线空间表示流形的局部几何形状，该切线空间是通过在每个数据点的邻域中拟合仿射子空间而学习的。通过邻域连接矩阵的部分本征分解，将这些切线对齐以给出数据点相对于基础流形的内部全局坐标。我们对算法进行了仔细的误差分析，表明构造误差具有二阶精度。我们使用2D / 3D和更高维的欧式空间中的曲线和曲面，以及具有各种姿势和光照条件的64 x 64像素脸部图像来说明我们的算法。我们还将解决一些理论和算法问题，以供进一步研究和改进。
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