Matrix factorization is a popular low dimensional representation approach that plays an important role in many pattern recognition and computer vision domains. Among them, convex and semi-nonnegative matrix factorizations have attracted considerable interest, owing to its clustering interpretation. On the other hand, the generalized correlation function (correntropy) as the error measure does not depend on the assumption of Gaussianity, which the mean square error (MSE) heavily depends on. In this paper, we propose two novel algorithms, called Maximum Correntropy Criterion based Convex and Semi-Nonnegative Matrix Factorization (MCC-ConvexNMF, MCCSemiNMF). Compared with the mean square error based convex and semi-nonnegative matrix factorization, the proposed methods can extract more information from the data and produce more accurate solutions. Experimental results on both synthetic dataset and the popular face database illustrate the effectiveness of our methods.