Issue No. 09 - September (2009 vol. 21)
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TKDE.2008.204
Shiming Xiang , Tsinghua National Laboratory for Information Science and Technology, Beijing
Feiping Nie , Tsinghua National Laboratory for Information Science and Technology, Beijing
Changshui Zhang , Tsinghua National Laboratory for Information Science and Technology, Beijing
Chunxia Zhang , Beijing Institute of Computer Science, Beijing
This paper presents a new algorithm for Nonlinear Dimensionality Reduction (NLDR). Our algorithm is developed under the conceptual framework of compatible mapping. Each such mapping is a compound of a tangent space projection and a group of splines. Tangent space projection is estimated at each data point on the manifold, through which the data point itself and its neighbors are represented in tangent space with local coordinates. Splines are then constructed to guarantee that each of the local coordinates can be mapped to its own single global coordinate with respect to the underlying manifold. Thus, the compatibility between local alignments is ensured. In such a work setting, we develop an optimization framework based on reconstruction error analysis, which can yield a global optimum. The proposed algorithm is also extended to embed out of samples via spline interpolation. Experiments on toy data sets and real-world data sets illustrate the validity of our method.
Nonlinear dimensionality reduction, compatible mapping, local spline embedding, out of samples.
C. Zhang, C. Zhang, S. Xiang and F. Nie, "Nonlinear Dimensionality Reduction with Local Spline Embedding," in IEEE Transactions on Knowledge & Data Engineering, vol. 21, no. , pp. 1285-1298, 2008.