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Spectral Sparse Representation for Clustering: Evolved from PCA, K-means, Laplacian Eigenmap, and Ratio Cut

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Abstract:

Dimensionality reduction methods, e.g. PCA and Laplacian eigenmap (LE), and cluster analysis methods, e.g. K-means and ratio cut (Rcut), are two kinds of widely-used unsupervised data analysis tools. The former concerns high representation fidelity, while the latter semantics. Some preliminary relations between these methods have been established in the literature, but they are not yet integrated into a unified framework. In this paper, we show that under an ideal condition, the four methods: PCA, K-means, LE, and Rcut, are unified together; and when the ideal condition is relaxed, the unification evolves to a new sparse representation method, called spectral sparse representation (SSR). It achieves the same representation fidelity as PCA/LE, and is able to reveal the cluster structure of data as K-means/Rcut. SSR is inherently related to cluster analysis, and the sparse codes can be directly used to do clustering. An efficient algorithm NSCrt is developed to solve the sparse codes of SSR. It is observed that NSCrt is able to effectively recover the underlying solutions. As a direct application of SSR, a new clustering algorithm Scut is devised. It reaches the start-of-the-art performance among spectral clustering methods. Compared with K-means based clustering methods, Scut does not depend on initialization and avoids getting trapped in local minima; and the solutions are comparable to the optimal ones of K-means run many times. Extensive experiments using data sets of different nature demonstrate the properties and strengths of SSR, NSCrt, and Scut.

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