Regularized matrix completion with partial side information

Abstract Side information has been shown useful for improving the performance of matrix completion applications. However, in most cases, only partial side information of either the column or row space is available. In this work, we propose a novel regularization based model to incorporate partial side information in matrix completion. We provide theoretical guarantees to ensure the success of the proposed model. It is proved that the proposed model achieves the state-of-the-art sample complexity when the given partial side information is exact, and an error bound for inexact partial side information is also provided. Moreover, we provide a deterministic rule for the selection of regularization parameter. We conduct extensive experiments on both synthetic and real-world data-sets. Experimental results show that our model succeeds to incorporate partial side information, and outperforms the state-of-the-art models on most data-sets.

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

[2]  Andrea Montanari,et al.  Matrix Completion from Noisy Entries , 2009, J. Mach. Learn. Res..

[3]  Sachin Garg,et al.  Response prediction using collaborative filtering with hierarchies and side-information , 2011, KDD.

[4]  Sujay Sanghavi,et al.  Completing any low-rank matrix, provably , 2013, J. Mach. Learn. Res..

[5]  Raymond K. W. Wong,et al.  Matrix Completion With Covariate Information , 2018, Journal of the American Statistical Association.

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

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

[8]  Robert D. Nowak,et al.  Transduction with Matrix Completion: Three Birds with One Stone , 2010, NIPS.

[9]  Nagarajan Natarajan,et al.  Inductive matrix completion for predicting gene–disease associations , 2014, Bioinform..

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

[11]  Min-Ling Zhang,et al.  A Review on Multi-Label Learning Algorithms , 2014, IEEE Transactions on Knowledge and Data Engineering.

[12]  G. Watson.  Characterization of the subdifferential of some matrix norms , 1992 .

[13]  Yudong Chen,et al.  Incoherence-Optimal Matrix Completion , 2013, IEEE Transactions on Information Theory.

[14]  Akshay Krishnamurthy,et al.  Low-Rank Matrix and Tensor Completion via Adaptive Sampling , 2013, NIPS.

[15]  Alexandre Bernardino,et al.  Matrix Completion for Weakly-Supervised Multi-Label Image Classification , 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[17]  Lei Yao,et al.  Partial Matrix Completion Algorithm for Efficient Data Gathering in Wireless Sensor Networks , 2015, IEEE Communications Letters.

[18]  Nasser M. Nasrabadi,et al.  Pattern Recognition and Machine Learning , 2006, Technometrics.

[19]  Deng Cai,et al.  Depth Image Inpainting: Improving Low Rank Matrix Completion With Low Gradient Regularization , 2017, IEEE Transactions on Image Processing.

[20]  Jicong Fan,et al.  Matrix completion by least-square, low-rank, and sparse self-representations , 2017, Pattern Recognit..

[21]  Dehui Yang,et al.  Weighted Matrix Completion and Recovery With Prior Subspace Information , 2018, IEEE Transactions on Information Theory.

[22]  Xue Liu,et al.  Data loss and reconstruction in sensor networks , 2013, 2013 Proceedings IEEE INFOCOM.

[23]  Zhi-Hua Zhou,et al.  ML-KNN: A lazy learning approach to multi-label learning , 2007, Pattern Recognit..

[24]  Naonori Ueda,et al.  Parametric Mixture Models for Multi-Labeled Text , 2002, NIPS.

[25]  F. Maxwell Harper,et al.  The MovieLens Datasets: History and Context , 2016, TIIS.

[26]  Angshul Majumdar,et al.  Matrix completion incorporating auxiliary information for recommender system design , 2015, Expert Syst. Appl..

[27]  Lei Yao,et al.  A DCT Regularized Matrix Completion Algorithm for Energy Efficient Data Gathering in Wireless Sensor Networks , 2015, Int. J. Distributed Sens. Networks.

[28]  Daniel Rueckert,et al.  Segmentation of MR images via discriminative dictionary learning and sparse coding: Application to hippocampus labeling , 2013, NeuroImage.

[29]  Walter Willinger,et al.  Spatio-temporal compressive sensing and internet traffic matrices , 2009, SIGCOMM '09.

[30]  Jicong Fan,et al.  Non-linear matrix completion , 2017, Pattern Recognit..

[31]  Inderjit S. Dhillon,et al.  Tumblr Blog Recommendation with Boosted Inductive Matrix Completion , 2015, CIKM.

[32]  Yueting Zhuang,et al.  Exploiting temporal stability and low-rank structure for motion capture data refinement , 2014, Inf. Sci..

[33]  David P. Woodruff,et al.  Matrix Completion and Related Problems via Strong Duality , 2017, ITCS.

[34]  Nigel Boston,et al.  A characterization of deterministic sampling patterns for low-rank matrix completion , 2015, 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[35]  Geng Yang,et al.  Anomaly-Tolerant Traffic Matrix Estimation via Prior Information Guided Matrix Completion , 2017, IEEE Access.

[36]  Miao Xu,et al.  Speedup Matrix Completion with Side Information: Application to Multi-Label Learning , 2013, NIPS.

[37]  Y. Zhang,et al.  Augmented Lagrangian alternating direction method for matrix separation based on low-rank factorization , 2014, Optim. Methods Softw..

[38]  Andrea Montanari,et al.  Matrix completion from a few entries , 2009, 2009 IEEE International Symposium on Information Theory.