  This paper provides a theoretical support for clustering aspect of the
nonnegative matrix factorization (NMF). By utilizing the Karush-Kuhn-Tucker
optimality conditions, we show that NMF objective is equivalent to graph
clustering objective, so clustering aspect of the NMF has a solid
justification. Different from previous approaches which usually discard the
nonnegativity constraints, our approach guarantees the stationary point being
used in deriving the equivalence is located on the feasible region in the
nonnegative orthant. Additionally, since clustering capability of a matrix
decomposition technique can sometimes imply its latent semantic indexing (LSI)
aspect, we will also evaluate LSI aspect of the NMF by showing its capability
in solving the synonymy and polysemy problems in synthetic datasets. And more
extensive evaluation will be conducted by comparing LSI performances of the NMF
and the singular value decomposition (SVD), the standard LSI method, using some
standard datasets.
