Semi-Supervised Classification via Local Spline Regression

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 32; no. 11; pp. 2039 - 2053
• Main Authors: Xiang, Shiming, Nie, Feiping, Zhang, Changshui
• Format: Journal Article
• Language: English
• Published: Los Alamitos, CA IEEE 01.11.2010; IEEE Computer Society; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Animals; Applied sciences; Artificial Intelligence; Classification; Cluster Analysis; Computer science; control theory; systems; Data points; Data processing. List processing. Character string processing; Databases, Factual; Exact sciences and technology; Humans; Image Processing, Computer-Assisted - methods; Image segmentation; interactive image segmentation; Iterative algorithms; Labeling; Labels; Laplace equations; Least-Squares Analysis; local spline regression; Machine learning; Machine learning algorithms; Memory organisation. Data processing; Optimization; Pattern Recognition, Automated - methods; Pattern recognition. Digital image processing. Computational geometry; Polynomials; Regression; Semi-supervised classification; Semisupervised learning; Software; Spline; Splines; Studies; Web pages; local spline regression; Image processing; Sobolev space; High precision; Semi-supervised classification; Green function; Supervised classification; Laplacian; Spline approximation; Polynomial function; Image segmentation; interactive image segmentation; Supervised learning; Least squares method; Objective function; Learning algorithm; Pattern analysis; Artificial intelligence
• ISSN: 0162-8828; 1939-3539
• DOI: 10.1109/TPAMI.2010.35

This paper presents local spline regression for semi-supervised classification. The core idea in our approach is to introduce splines developed in Sobolev space to map the data points directly to be class labels. The spline is composed of polynomials and Green's functions. It is smooth, nonlinear, and able to interpolate the scattered data points with high accuracy. Specifically, in each neighborhood, an optimal spline is estimated via regularized least squares regression. With this spline, each of the neighboring data points is mapped to be a class label. Then, the regularized loss is evaluated and further formulated in terms of class label vector. Finally, all of the losses evaluated in local neighborhoods are accumulated together to measure the global consistency on the labeled and unlabeled data. To achieve the goal of semi-supervised classification, an objective function is constructed by combining together the global loss of the local spline regressions and the squared errors of the class labels of the labeled data. In this way, a transductive classification algorithm is developed in which a globally optimal classification can be finally obtained. In the semi-supervised learning setting, the proposed algorithm is analyzed and addressed into the Laplacian regularization framework. Comparative classification experiments on many public data sets and applications to interactive image segmentation and image matting illustrate the validity of our method.