Two Models For Orthogonal Nonnegative Matrix Factorization Based On L_p Optimization

Nonnegative matrix factorization(NMF)has many good properties and has been studied in various areas including models,algorithms and some researches of application.However,the first proposed NMF model is difficult to satisfy differ-ent demands in applications.So,many relative models for NMF problems were proposed.There is an orthogonal nonnegative matrix factorization model(ONMF)that adds orthogonal constrain on the NMF model to help interpret the results.Also,we can make use of the prior information about real situations that we ob-served before such as low rank,smoothness and sparsity.Then,add corresponding consrtains on the model to improve the performances in applications.For sparsity,a l1 norm is commonly used as a sparse constrain on a matrix instead of a to norm.But this is not the best way to achieve sparsity.A lp norm(0<p<1)is better than the l1 norm.Thus,extracting some latent properties and using a lot of relative academic knowledge to propose models will satisfy the demands in our applications.We proposed two new ONMF models.The first one is an orthogonal nonnega-tive matrix factorization model based on lp optimization(lp-ONMF).Although the lp norm(0<p<1)is a nonconvex function,it has better representation on spar-sity because it is closer to the l0 norm.Also,this nonconvex model is still solvable.Thus,we choose the lp norm as a sparse constrain in this model.The second one is a smooth orthogonal nonnegative matrix factorization model based on lp optimization(TV-lp-ONMF)which is on the basis of the first proposed model by adding total variation regularization to achieve smoothness.Alternating direction method of multipliers(ADMM)is used to find the optimal solutions in two new proposed models in the light of characteristics of them in this paper.Later,we apply two new models to actual data experiments such as text clustering and image segmentation.In text clustering,the lp-ONMF model outperforms many state-of-art methods that are tested in this paper.The TV-lp-ONMF model is better than that of all the methods compared in this paper,and the clustering performances of more than half of the tested text data sets are higher than 95%respectively.In image segmentation,two proposed methods can find the correct parts that are composed of the image.The experiments illustrate the feasibility of the two proposed models.