By incorporating the geometric structure, GNMF can significantly improve representation ability in comparison to conventional NMF.
In conventional GNMF, the intrinsic low dimensional manifold structure embedded in the dataset is expressed by n by n graph weight matrix W.
In this section, we verify the proposed PNMF-M by comparing it with PNMF in  and GNMF in , which incorporate parallel computing and manifold regularization into NMF, respectively.
Supervised label information is added to the objective function of GNMF .
In order to verify the effectiveness of GLD-RNMF algorithm for identifying differentially expressed genes, we perform experiments on real gene expression datasets to compare our algorithm with the other four feature extraction algorithms: (a) GNMF algorithm (Cai et al.
To be fair, we extract 100 genes from the gene expression data by GNMF, NMFSC, RGNMF, GDNMF, and GLDRNMF methods.
Experiment results demonstrate SPNMF outperforms GNMF and its variants.
The overall cost for GNMF is O(tMNK + M[N.sup.2]) and the overall cost for SPNMF is O(t(M + N)NK + M[N.sub.2]).
(iv) Graph regularized nonnegative matrix factorization based clustering with F-norm formulation (GNMF in short).
GNMF. The geometrical information of the original data space is an important information for face recognition [20, 21]; data set often lay in a high dimension manifold, so the intrinsic geodesic distance in the manifold between two data points is more suitable than Euclidean distance.
[x.sub.k]] [member of] [R.sup.nxk], there are total k face image vectors and [x.sub.i] (1 [less than or equal to] i [less than or equal to] k) denotes a vector of n-dimensional facial image, U [member of] [R.sup.nxr] and V [member of] [R.sup.rxk], so the objective function F(U,V) for GNMF is defined as (1); by minimizing F(U, V) a factorization, U and V in GNMF, is obtained:
Assuming [F.sup.k.sub.k+1] refers to the objective function corresponding to GNMF representation of the first k sample when the (k + 1)th sample arrives, [L.sup.k.sub.k+1] refers to the k x k dimensional matrix which equals the first k rows and first k columns of [L.sub.k+1], and r is a predefined parameter which indicates that the n-dimensional facial image vector maps in to an r-dimensional vector.