Nonnegative Tucker Decomposition

Nonnegative tensor factorization (NTF) is a recent multiway (multilinear) extension of nonnegative matrix factorization (NMF), where nonnegativity constraints are imposed on the CANDECOMP/PARAFAC model. In this paper we consider the Tucker model with nonnegativity constraints and develop a new tensor factorization method, referred to as nonnegative Tucker decomposition (NTD). The main contributions of this paper include: (1) multiplicative updating algorithms for NTD; (2) an initialization method for speeding up convergence; (3) a sparseness control method in tensor factorization. Through several computer vision examples, we show the useful behavior of the NTD, over existing NTF and NMF methods.

[1]  J. Leeuw,et al.  Principal component analysis of three-mode data by means of alternating least squares algorithms , 1980 .

[2]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[3]  Narendra Ahuja,et al.  Rank-R approximation of tensors using image-as-matrix representation , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[4]  H. Sebastian Seung,et al.  Learning the parts of objects by non-negative matrix factorization , 1999, Nature.

[5]  Max Welling,et al.  Positive tensor factorization , 2001, Pattern Recognit. Lett..

[6]  Demetri Terzopoulos,et al.  Multilinear Analysis of Image Ensembles: TensorFaces , 2002, ECCV.

[7]  P. Paatero,et al.  Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values† , 1994 .

[8]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[9]  Patrik O. Hoyer,et al.  Non-negative Matrix Factorization with Sparseness Constraints , 2004, J. Mach. Learn. Res..

[10]  Demetri Terzopoulos,et al.  Multilinear subspace analysis of image ensembles , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[11]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .

[12]  H. Kiers.  Towards a standardized notation and terminology in multiway analysis , 2000 .

[13]  Tamara G. Kolda,et al.  Categories and Subject Descriptors: G.4 [Mathematics of Computing]: Mathematical Software— , 2022 .

[14]  J. Kruskal.  Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics , 1977 .

[15]  Daoqiang Zhang,et al.  Two-Dimensional Non-negative Matrix Factorization for Face Representation and Recognition , 2005, AMFG.

[16]  M. Alex O. Vasilescu.  Multilinear independent component analysis , 2004 .

[17]  Richard A. Harshman,et al.  Determination and Proof of Minimum Uniqueness Conditions for PARAFAC1 , 1972 .

[18]  Tamir Hazan,et al.  Non-negative tensor factorization with applications to statistics and computer vision , 2005, ICML.

[19]  Demetri Terzopoulos,et al.  Multilinear independent components analysis , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[20]  Tamir Hazan,et al.  Sparse image coding using a 3D non-negative tensor factorization , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[21]  Alejandro F. Frangi,et al.  Two-dimensional PCA: a new approach to appearance-based face representation and recognition , 2004 .

[22]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[23]  Andy Harter,et al.  Parameterisation of a stochastic model for human face identification , 1994, Proceedings of 1994 IEEE Workshop on Applications of Computer Vision.

[24]  Andrzej Cichocki,et al.  Non-Negative Tensor Factorization using Alpha and Beta Divergences , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.