Improving nonnegative matrix factorization with advanced graph regularization☆
Section snippets
1 Introduction
A recommendation system is a vital component of the modern Internet environment. It can help people locate their favorite items and filter the desired information out of a huge amount of data [5], [29]. Collaborative Filtering (CF) [1], [2], [19], [34] is one of the most popular methods for recommendation systems, especially for the large scale systems including movies [4], music [21], television [3], books [32], e-learning [5] or e-commerce [8]. Assuming users who liked similar items in the
2 Preliminaries
In this section, we briefly introduce the basic definitions of NMF and GNMF and their update rules for factorization. Before introducing these methods, we provide some notations that will be used throughout in this paper.
Suppose that we have a rating matrix consisting of M users and N items (or movies). Let be the rating matrix (see Table 1), where is the jth column of . Indeed, is the rating vector for item j from all users, and
3. Linear Projection and Graph Regularized Nonnegative Matrix Factorization
NMF can learn a parts-based representation in the Euclidean space, but it is unable to unveil the inherent geometric structure of the original data space, which is important for practical applications. Although GNMF solved this problem by introducing a new regularizer with geometric structure, it still doesn’t involve the observed ratings of items nor the geometric structure of its projection matrix. In contrast, LPGNMF fuses three elements, user-item ratings matrix, item matrix and user
Experiments
In this section, we report several experiments to evaluate the performance of LPGNMF and F-LPGNMF and compare them with the existing approaches on eight data sets: Movielens 100 K, MovieLens 1 M, MovieLens 10 M, MovieLens-Latest-Small and Yelp data set (Pets, Active Life, Restaurants, and Beauty and Spas). The compared approaches include SR1VSS, SR2VSS, PRM2, NMF [24], [33], GNMF and GUNMF [6].
Performance comparison
We compare the performance of LPGNMF and F-LPGNMF with the existing methods including SR1VSS, SR2VSS [30], PRM2 [35], NMF [24], [33], GNMF and GUNMF [6] on two data sets: MovieLens and Yelp. Since the training set and testing set are divided randomly, and the setting of initial values of latent matrices is random, the results may slightly differ at each time. To solve this problem and reduce experimental error, we have repeated this procedure 20 times and taken their mean as the following
Conclusion
In this paper, we present a novel regularizer for nonnegative matrix factorization based on linear projection and geometric structure. We develop two iterative update formulas to minimize the proposed objective function, one is proposed based on multiplicative update rules and called LPGNMF, and another one is based alternative direction principle and called F-LPGNMF. The test results demonstrate LPGNMF and F-LPGNMF have a better prediction performance for movie ratings than SR1VSS, SR2VSS,
CRediT authorship contribution statement
Xiaoxia Zhang: Supervision, Investigation, Conceptualization, Methodology, Software, Writing - original draft, Validation. Degang Chen: Data curation, Resources, Formal analysis, Investigation. Hong Yu: Funding acquisition. Guoyin Wang: Funding acquisition. Kesheng Wu: Writing - review & editing, Visualization, Investigation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors would like to thank the National Natural Science Fund of China (61876027), the State Key Program of National Nature Science Foundation of China (61936001), the Science and Technology Research Program of Chongqing Education Commission of China (KJQN201900638) and the National Energy Research Scientific Computing Center (NERSC).
References (41)
- et al.
Collaborative filtering adapted to recommender systems of e-learning
Knowl.-Based Syst.
(2009) - et al.
A non negative matrix factorization for collaborative filtering recommender systems based on a Bayesian probabilistic model
Knowl.-Based Syst.
(2016) - et al.
Collaborative filtering with ordinal scale-based implicit ratings for mobile music recommendations
Inf. Sci.
(2010) - et al.
Graph dual regularization non-negative matrix factorization for co-clustering
Pattern Recogn.
(2012) - et al.
Compressed knowledge transfer via factorization machine for heterogeneous collaborative recommendation
Knowl.-Based Syst.
(2015) - et al.
NMFE-SSCC: Non-negative matrix factorization ensemble for semi-supervised collective classification
Knowl.-Based Syst.
(2015) - et al.
Improving aggregate recommendation diversity using ranking-based techniques
IEEE Trans. Knowl. Data Eng.
(2012) - et al.
Toward the next generation of recommender systems: A survey of the state-of-the-art and possible extensions
IEEE Trans. Knowl. Data Eng.
(2005) - et al.
A hybrid content-based and item-based collaborative filtering approach to recommend TV programs enhanced with singular value decomposition
Inf. Sci.
(2010) - et al.
Lessons from the Netflix prize challenge
SIGKDD Explor.
(2007)
Graph regularized nonnegative matrix factorization for data representation
IEEE Trans. Pattern Anal. Mach. Intell.
LPGNMF: predicting long non-coding rna and protein interaction using graph regularized nonnegative matrix factorization
IEEE/ACM Trans. Comput. Biol. Bioinform.
A highly adaptive recommender system based on fuzzy logic for b2c e-commerce portals
Expert Syst. Appl.
Collaborative filtering using orthogonal nonnegative matrix tri-factorization, Data Mining Workshops, 2007 ICDM Workshops 2007
Seventh IEEE International Conference on IEEE
Spectral Graph Theory
Am. Math. Soc.
Maximum likelihood from incomplete data via the Em algorithm
J. R. Stat. Soc. Ser. B (Methodological)
Convex and semi-nonnegative matrix-factorizations
IEEE Trans. Pattern Anal. Mach. Intell.
Scalable recommendations with poisson factorization
Eprint Arxiv
Online nonnegative matrix factorization with robust stochastic approximation
IEEE Trans. Neural Networks Learn. Syst.
Accelerating the Lee-Seung algorithm for non-negative matrix factorization
Cited by (0)
- ☆
This work is jointly supported by the National Key R & D Program of China (2019YFB2103000), the National Natural Science Foundation of China (61936001, 62136002, 61876027), the Natural Science Foundation of Chongqing (cstc2019jcyj-cxttX0002, cstc2020jcyj-msxmX0737, cstc2021ycjh-bgzxm0013), the Key Cooperation Project of Chongqing Municipal Education Commission (HZ2021008), the Science and Technology Research Program of Chongqing Education Commission of China (KJQN201900638), the National Energy Research Scientific Computing Center (NERSC).