Sparsity-Aware Possibilistic Clustering Algorithms

Abstract

In this paper two novel possibilistic clustering algorithms are presented, which utilize the concept of sparsity. The first one, called sparse possibilistic c-means, exploits sparsity and can deal well with closely located clusters that may also be of significantly different densities. The second one, called sparse adaptive possibilistic c-means, is an extension of the first, where now the involved parameters are dynamically adapted. The latter can deal well with even more challenging cases, where, in addition to the above, clusters may be of significantly different variances. More specifically, it provides improved estimates of the cluster representatives, while, in addition, it has the ability to estimate the actual number of clusters, given an overestimate of it. Extensive experimental results on both synthetic and real data sets support the previous statements.

An extended Takagi–Sugeno–Kang inference system (TSK+) with fuzzy interpolation and its rule base generation

“…Firstly, the sparsity-aware possibilistic clustering algorithm (Xenaki et al (2016)) was designed to work with sparse data sets, and thus, it is desired to investigate how this clustering algorithm can support the proposed TSK+ inference system. Secondly, it is worthwhile to study how the proposed sparse rule base generation approach can help in generating Mamdani-style fuzzy rule bases.…”

On the Convergence of the Sparse Possibilistic C-Means Algorithm

“…Note, however, that the actual complexity is much less since at each iteration the bisection method is activated only for a small fraction of u ij 's. As it is shown experimentally in [16] the computational complexity of SPCM is slightly increased compared to that of PCM. This is the price to pay for the better quality results of SPCM compared to PCM.…”

“…Taking into account the definition of h(u i ; θ) in eq. (24), it can be shown (Proposition 5, [16]) that the maximum of the two solutions u…”

“…ij , the largest of which corresponds to a local minimum of J SP CM with respect to u ij . In [16] it is shown that J SP CM (U, Θ) exhibits its global minimum at u * ij , where:…”

“…2 A different approach for proving the convergence of the FCM to a stationary point of the corresponding cost function is given in [14]. A relative work is also provided in [15].August 6, 2018 DRAFT (SPCM) [16], has been proposed, which extends PCM 2 by introducing sparsity. More specifically, a suitable sparsity constraint is imposed on the vectors containing the degrees of compatibility of the data points with the clusters (one vector per point 3 ), such that each data vector is compatible with only a few or even none clusters.…”