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In this paper, we present a novel algorithm for representing facial expressions. The algorithm is based on the non-negative matrix factorization (NMF) algorithm, which decomposes the original facial image matrix into two non-negative matrices, namely the coefficient matrix and the basis image matrix. We call the novel algorithm graph-preserving sparse non-negative matrix factorization (GSNMF). GSNMF utilizes both sparse and graph-preserving constraints to achieve a non-negative factorization. The graph-preserving criterion preserves the structure of the original facial images in the embedded subspace while considering the class information of the facial images. Therefore, GSNMF has more discriminant power than NMF. GSNMF is applied to facial images for the recognition of six basic facial expressions. Our experiments show that GSNMF achieves on average a recognition rate of 93.5% compared to that of discriminant NMF with 91.6%.