Clustering and latent semantic indexing aspects of the non-negative matrix factorisation

This paper proposes a theoretical support for clustering aspect of non-negative matrix factorisation (NMF). By utilising Karush-Kuhn-Tucker optimality conditions, we show that NMF objective is equivalent to graph clustering objective, so clustering aspect of NMF has a solid justification. Different from previous approaches - which either ignore non-negativity constraints or assume absolute orthonormality on coefficient matrix in order to derive the equivalency - our approach takes non-negativity constraints into account and makes no assumption about orthonormality of coefficient matrix. Thus, not only stationary point being used in deriving the equivalency is guaranteed to be located on NMFs feasible region, but also the result is more realistic since NMF does not produce orthonormal matrix. Furthermore, because clustering capability of a matrix decomposition technique may imply its latent semantic indexing (LSI) aspect, we also study LSI aspect of NMF.