Alternating proximal gradient method for sparse nonnegative Tucker decomposition

Yangyang Xu

Research output: Contribution to journalArticlepeer-review

54 Scopus citations


Multi-way data arises in many applications such as electroencephalography classification, face recognition, text mining and hyperspectral data analysis. Tensor decomposition has been commonly used to find the hidden factors and elicit the intrinsic structures of the multi-way data. This paper considers sparse nonnegative Tucker decomposition (NTD), which is to decompose a given tensor into the product of a core tensor and several factor matrices with sparsity and nonnegativity constraints. An alternating proximal gradient method is applied to solve the problem. The algorithm is then modified to sparse NTD with missing values. Per-iteration cost of the algorithm is estimated scalable about the data size, and global convergence is established under fairly loose conditions. Numerical experiments on both synthetic and real world data demonstrate its superiority over a few state-of-the-art methods for (sparse) NTD from partial and/or full observations. The MATLAB code along with demos are accessible from the author’s homepage.

Original languageEnglish (US)
Pages (from-to)39-70
Number of pages32
JournalMathematical Programming Computation
Issue number1
StatePublished - Mar 2015
Externally publishedYes

Bibliographical note

Funding Information:
This work is partly supported by ARL and ARO grant W911NF-09-1-0383 and AFOSR FA9550-10-C-0108. The author would like to thank three anonymous referees, the technical editor and the associate editor for their very valuable comments and suggestions. Also, the author would like to thank Prof. Wotao Yin for his valuable discussions and Anh Huy Phan for sharing the code of HALS.

Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.


  • Alternating proximal gradient method
  • Non-convex optimization
  • Sparse nonnegative Tucker decomposition
  • Sparse optimization


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