Fuzzy c-means is one of the popular algorithms in clustering, but it has some drawbacks such as sensitivity to outliers. Although many correntropy based works have been proposed to improve the robustness of FCM, fundamentally a proper error function is required to apply to FCM. In this paper, we present a new perspective based on the expected loss (or risk) to FCM method to provide different kinds of robustness such as robustness to outliers, to the volume of clusters and robustness in noisy environments. First, we propose Robust FCM method (RCM) by defining a loss function as a least square problem and benefiting the correntropy to make FCM robust to outliers. Furthermore, we utilize the half-quadratic (HQ) optimization as a problem-solving method. Second, inspiring by the Bayesian perspective, we define a new loss function based on correntropy as a distance metric to present Robust Heterogeneous C-Means (RHCM) by utilizing direct clustering (DC) method. DC helps RHCM to have robust initialization. Besides, RHCM will make some robust cluster centers in noisy environments and is capable of clustering the elliptical or spherical shaped data accurately, regardless of the volume of each cluster. The results are shown visually on some synthetic datasets including the noisy ones, the UCI repository and also on real image dataset that was gathered manually from 500px social media. Also, for evaluation of the clustering results, several validity indices are calculated. Experimental results indicate the superiority of our proposed method over the base FCM, DC, KFCM, two new methods called GPFCM and GEPFCM and a method called DC-KFCM that we created for the comparison purpose.

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