Local Laplacian Coding From Theoretical Analysis of Local Coding Schemes for Locally Linear Classification

• Published in: IEEE transactions on cybernetics Vol. 45; no. 12; pp. 2937 - 2947
• Main Authors: Pang, Junbiao, Qin, Lei, Zhang, Chunjie, Zhang, Weigang, Huang, Qingming, Yin, Baocai
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.12.2015; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Approximation; Classification; Coding; Cybernetics; Encoding; Flexibility; Gaussian; Image classification; Laplace equations; Linear approximation; local coordinate coding (LCC); local Gaussian coding (LGC); local Laplacian coding (LPC); local student coding (LSC); locally linear classification; Manifolds; Mathematical analysis; nonlinear approximation; Nonlinearity; Symmetric matrices; Tasks; local Gaussian coding (LGC); local coordinate coding (LCC); nonlinear approximation; locally linear classification; local Laplacian coding (LPC); Image classification; local student coding (LSC)
• ISSN: 2168-2267; 2168-2275
• DOI: 10.1109/TCYB.2015.2433926

Local coordinate coding (LCC) is a framework to approximate a Lipschitz smooth function by combining linear functions into a nonlinear one. For locally linear classification, LCC requires a coding scheme that heavily determines the nonlinear approximation ability, posing two main challenges: 1) the locality making faraway anchors have smaller influences on current data and 2) the flexibility balancing well between the reconstruction of current data and the locality. In this paper, we address the problem from the theoretical analysis of the simplest local coding schemes, i.e., local Gaussian coding and local student coding, and propose local Laplacian coding (LPC) to achieve the locality and the flexibility. We apply LPC into locally linear classifiers to solve diverse classification tasks. The comparable or exceeded performances of state-of-the-art methods demonstrate the effectiveness of the proposed method.