Nonlinear Dimensionality Reduction with Local Spline Embedding

• Published in: IEEE transactions on knowledge and data engineering Vol. 21; no. 9; pp. 1285 - 1298
• Main Authors: Xiang, Shiming, Nie, Feiping, Zhang, Changshui, Zhang, Chunxia
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2009; IEEE Computer Society; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Artificial intelligence; compatible mapping; Computer science; control theory; systems; Computer vision; Data mining; Data points; Data processing. List processing. Character string processing; Data visualization; Error analysis; Exact sciences and technology; Image reconstruction; Interpolation; local spline embedding; Manifolds; Memory organisation. Data processing; Nonlinear dimensionality reduction; Optimization; out of samples; Pattern recognition; Principal component analysis; Projection; Software; Spline; Splines; Studies; Tangents; General; Pattern analysis; Machine learning; Dimension reduction; Alignment; Error estimation; compatible mapping; out of samples; Global optimum; local spline embedding; Compatibility; Optimization; Spline approximation; Nonlinear dimensionality reduction
• ISSN: 1041-4347; 1558-2191
• DOI: 10.1109/TKDE.2008.204

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.