Joint Schatten $$p$$p-norm and $$\ell _p$$ℓp-norm robust matrix completion for missing value recovery

The low-rank matrix completion problem is a fundamental machine learning and data mining problem with many important applications. The standard low-rank matrix completion methods relax the rank minimization problem by the trace norm minimization. However, this relaxation may make the solution seriously deviate from the original solution. Meanwhile, most completion methods minimize the squared prediction errors on the observed entries, which is sensitive to outliers. In this paper, we propose a new robust matrix completion method to address these two problems. The joint Schatten $$p$$p-norm and $$\ell _p$$ℓp-norm are used to better approximate the rank minimization problem and enhance the robustness to outliers. The extensive experiments are performed on both synthetic data and real-world applications in collaborative filtering prediction and social network link recovery. All empirical results show that our new method outperforms the standard matrix completion methods.

[1]  Michael J. Pazzani,et al.  Learning Collaborative Information Filters , 1998, ICML.

[2]  Benjamin Recht,et al.  A Simpler Approach to Matrix Completion , 2009, J. Mach. Learn. Res..

[3]  Stephen P. Boyd,et al.  A rank minimization heuristic with application to minimum order system approximation , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[4]  Jieping Ye,et al.  An accelerated gradient method for trace norm minimization , 2009, ICML '09.

[5]  S. Yun,et al.  An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems , 2009 .

[6]  Robert Tibshirani,et al.  Spectral Regularization Algorithms for Learning Large Incomplete Matrices , 2010, J. Mach. Learn. Res..

[7]  Francis R. Bach,et al.  A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization , 2008, J. Mach. Learn. Res..

[8]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[9]  Sriram Vishwanath,et al.  Information theoretic bounds for low-rank matrix completion , 2010, 2010 IEEE International Symposium on Information Theory.

[10]  Feiping Nie,et al.  Social Trust Prediction Using Rank-k Matrix Recovery , 2013, IJCAI.

[11]  M. J. D. Powell,et al.  A method for nonlinear constraints in minimization problems , 1969 .

[12]  V. Koltchinskii,et al.  Nuclear norm penalization and optimal rates for noisy low rank matrix completion , 2010, 1011.6256.

[13]  Jure Leskovec,et al.  Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters , 2008, Internet Math..

[14]  Yong-Jin Liu,et al.  An implementable proximal point algorithmic framework for nuclear norm minimization , 2012, Math. Program..

[15]  Pablo A. Parrilo,et al.  Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , 2007, SIAM Rev..

[16]  M. Newman.  Clustering and preferential attachment in growing networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Shiqian Ma,et al.  Fixed point and Bregman iterative methods for matrix rank minimization , 2009, Math. Program..

[18]  Emmanuel J. Candès,et al.  A Singular Value Thresholding Algorithm for Matrix Completion , 2008, SIAM J. Optim..

[19]  B. Mercier,et al.  A dual algorithm for the solution of nonlinear variational problems via finite element approximation , 1976 .

[20]  Nathan Srebro,et al.  Fast maximum margin matrix factorization for collaborative prediction , 2005, ICML.

[21]  Feiping Nie,et al.  Low-Rank Matrix Recovery via Efficient Schatten p-Norm Minimization , 2012, AAAI.

[22]  John Wright,et al.  Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization , 2009, NIPS.

[23]  Emmanuel J. Candès,et al.  The Power of Convex Relaxation: Near-Optimal Matrix Completion , 2009, IEEE Transactions on Information Theory.

[24]  Feiping Nie,et al.  Robust Matrix Completion via Joint Schatten p-Norm and lp-Norm Minimization , 2012, 2012 IEEE 12th International Conference on Data Mining.

[25]  Tommi S. Jaakkola,et al.  Maximum-Margin Matrix Factorization , 2004, NIPS.

[26]  Stephen P. Boyd,et al.  Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices , 2003, Proceedings of the 2003 American Control Conference, 2003..

[27]  Katja Hose,et al.  Partout: a distributed engine for efficient RDF processing , 2012, WWW.

[28]  Feiping Nie,et al.  Robust Discrete Matrix Completion , 2013, AAAI.

[29]  Vincent Y. F. Tan,et al.  Rank minimization over finite fields , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[30]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[31]  Chris H. Q. Ding,et al.  Collaborative Filtering: Weighted Nonnegative Matrix Factorization Incorporating User and Item Graphs , 2010, SDM.

[32]  Ruslan Salakhutdinov,et al.  Probabilistic Matrix Factorization , 2007, NIPS.

[33]  Richard M. Karp,et al.  The rank of sparse random matrices over finite fields , 1997 .

[34]  John Wright,et al.  Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization , 2009, NIPS.

[35]  Pierre Loidreau,et al.  Properties of codes in rank metric , 2006, ArXiv.

[36]  Inderjit S. Dhillon,et al.  Matrix Completion with Noisy Side Information , 2015, NIPS.

[37]  Ruslan Salakhutdinov,et al.  Collaborative Filtering in a Non-Uniform World: Learning with the Weighted Trace Norm , 2010, NIPS.

[38]  Inderjit S. Dhillon,et al.  Guaranteed Rank Minimization via Singular Value Projection , 2009, NIPS.

[39]  Feiping Nie,et al.  Efficient and Robust Feature Selection via Joint ℓ2, 1-Norms Minimization , 2010, NIPS.

[40]  Jure Leskovec,et al.  Predicting positive and negative links in online social networks , 2010, WWW '10.

[41]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2009, Found. Comput. Math..

[42]  Paul Tseng,et al.  Trace Norm Regularization: Reformulations, Algorithms, and Multi-Task Learning , 2010, SIAM J. Optim..