## Non-negative matrix factorisation based on fuzzy *K* nearest neighbour graph and its applications

Non-negative matrix factorisation (NMF) has been widely used in pattern recognition problems. For the tasks of classification, however, most of the existing variants of NMF ignore both the discriminative information and the local geometry of data into the factorisation. The actual conditions of the problems will be affected by the change of the environmental factors to affect the recognition accuracy. In order to overcome these drawbacks, the authors regularised NMF by intra-class and inter-class fuzzy *K* nearest neighbour graphs, leading to NMF-F*K*-NN in this study. By introducing two novel fuzzy *K* nearest neighbour graphs, NMF-F*K*-NN can contract the intra-class neighbourhoods and expand the inter-class neighbourhoods in the decomposition. This method not only exploits the discriminative information and uses the geometric structure in the data effectively, but also reduces the influence of the external factors to improve recognition effect. In the factorisation, the authors minimised the approximation error whilst contracting intra-class fuzzy neighbourhoods and expanding inter-class fuzzy neighbourhoods. The authors develop simple multiplicative updates for NMF-F*K*-NN and present monotonic convergence results. Experiments of the text clustering on the CLUTO toolkit and face recognition on ORL and YALE datasets show the effectiveness of our proposed method.