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Inverse Problems and Imaging (IPI)
 

Subspace clustering by $(k,k)$-sparse matrix factorization

Pages: 539 - 551, Volume 11, Issue 3, June 2017      doi:10.3934/ipi.2017025

 
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Haixia Liu - Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (email)
Jian-Feng Cai - Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (email)
Yang Wang - Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (email)

Abstract: High-dimensional data often lie in low-dimensional subspaces instead of the whole space. Subspace clustering is a problem to analyze data that are from multiple low-dimensional subspaces and cluster them into the corresponding subspaces. In this work, we propose a $(k,k)$-sparse matrix factorization method for subspace clustering. In this method, data itself is considered as the ``dictionary'', and each data point is represented as a linear combination of the basis of its cluster in the dictionary. Thus, the coefficient matrix is low-rank and sparse. With an appropriate permutation, it is also blockwise with each block corresponding to a cluster. With an assumption that each block is no more than $k$-by-$k$ in matrix recovery, we seek a low-rank and $(k,k)$-sparse coefficient matrix, which will be used for the construction of affinity matrix in spectral clustering. The advantage of our proposed method is that we recover a coefficient matrix with $(k,k)$-sparse and low-rank simultaneously, which is better fit for subspace clustering. Numerical results illustrate the effectiveness that it is better than SSC and LRR in real-world classification problems such as face clustering and motion segmentation.

Keywords:  Subspace clustering, matrix factorization, ($k,k$)-sparse, low-rank, $k$-support norm.
Mathematics Subject Classification:  Primary: 68T04; Secondary: 65F04.

Received: April 2016;      Revised: February 2017;      Available Online: April 2017.

 References