Fuzzy Clustering of Single-View Incomplete Data Using a Multiview Framework

We propose four frameworks for clustering data with missing values. We first use a very simple method to impute the missing values and generate multiple imputed versions of the data. These views are then clustered together to obtain a common partition matrix and a common set of centroids. As the clustering framework, we use a multiview version of the Fuzzy-c-Means (MVFCM) and a multiview version of Kernelized Fuzzy-c-Means (MVKFCM). To find the importance (weights) of different views, we use an entropic regularization term using the weights. After obtaining the optimal weights, the final imputation is done as a weighted sum (convex combination) of the imputed values used to generate the views. The final clustering is done on this imputed data set. We compare the performance of the proposed algorithms with several algorithms using Normalized Mutual Information, Adjusted Rand Index, and cluster accuracy on 12 benchmark data sets. Of these algorithms, the MVKFCM is found to perform the best. The MVFCM and MVKFCM use <inline-formula><tex-math notation="LaTeX">$5 \times r$</tex-math></inline-formula> views, where <inline-formula><tex-math notation="LaTeX">$r$</tex-math></inline-formula> is the number of classes (note that the class labels are not used). However, <inline-formula><tex-math notation="LaTeX">$r$</tex-math></inline-formula> may not be known, and also for large <inline-formula><tex-math notation="LaTeX">$r$</tex-math></inline-formula>, there will be too many views. So we propose two variants of MVKFCM: <inline-formula><tex-math notation="LaTeX">$MVKFCM_{FV}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$MVKFCM_{RFV}$</tex-math></inline-formula> (<italic>FV</italic> stands for a fixed number of views and <italic>RFV</italic> stands for a robust version with fixed views). The <inline-formula><tex-math notation="LaTeX">$MVKFCM_{RFV}$</tex-math></inline-formula> generates views in a manner that helps to obtain robust performance. As expected, <inline-formula><tex-math notation="LaTeX">$MVKFCM_{RFV}$</tex-math></inline-formula> is found to be the best performing algorithm.