Two-Dimensional Embedded Fuzzy Data Clustering

Abstract

Fuzzy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -means (FKM) is a popular clustering method by assigning data points into respective clusters with uncertainty measured by the membership degree. Usually, FKM performs clustering according to the distance between data points in the original space, which might contain undesirable noises and redundant features; therefore, the underlying data semantic connections cannot be accurately captured. Moreover, the vectorized representation of the two-dimensional data such as image inevitably leads to the loss of structural information. In this paper, we propose a novel FKM method termed two-dimensional embedded fuzzy data clustering (2DEFC) which has two merits. First, 2DEFC directly takes 2D data as input without vectorizing them in order to retain more structural information of data. Second, the two subspace projection matrices are jointly optimized with the data membership degree for better collaborating with each other, which effectively avoids the sub-optimality limitation caused by the conventional mode of sequentially performing dimensionality reduction and clustering. An efficient algorithm is proposed to optimize the 2DEFC objective function. Besides, we provide comprehensive analysis on 2DEFC including its convergence behavior, computational complexity, and the fuzzy weighting exponent. Extensive comparative studies on benchmark 2D data sets demonstrate that the competitive performance of 2DEFC in 2D data clustering.