\(\ell ^{0}\)-Sparse Subspace Clustering

  • Yingzhen YangEmail author
  • Jiashi Feng
  • Nebojsa Jojic
  • Jianchao Yang
  • Thomas S. Huang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9906)


Subspace clustering methods with sparsity prior, such as Sparse Subspace Clustering (SSC) [1], are effective in partitioning the data that lie in a union of subspaces. Most of those methods require certain assumptions, e.g. independence or disjointness, on the subspaces. These assumptions are not guaranteed to hold in practice and they limit the application of existing sparse subspace clustering methods. In this paper, we propose \(\ell ^{0}\)-induced sparse subspace clustering (\(\ell ^{0}\)-SSC). In contrast to the required assumptions, such as independence or disjointness, on subspaces for most existing sparse subspace clustering methods, we prove that subspace-sparse representation, a key element in subspace clustering, can be obtained by \(\ell ^{0}\)-SSC for arbitrary distinct underlying subspaces almost surely under the mild i.i.d. assumption on the data generation. We also present the “no free lunch” theorem that obtaining the subspace representation under our general assumptions can not be much computationally cheaper than solving the corresponding \(\ell ^{0}\) problem of \(\ell ^{0}\)-SSC. We develop a novel approximate algorithm named Approximate \(\ell ^{0}\)-SSC (\(\hbox {A}\ell ^{0}\)-SSC) that employs proximal gradient descent to obtain a sub-optimal solution to the optimization problem of \(\ell ^{0}\)-SSC with theoretical guarantee, and the sub-optimal solution is used to build a sparse similarity matrix for clustering. Extensive experimental results on various data sets demonstrate the superiority of \(\hbox {A}\ell ^{0}\)-SSC compared to other competing clustering methods.


Sparse subspace clustering Proximal gradient descent 

Supplementary material

419974_1_En_45_MOESM1_ESM.pdf (153 kb)
Supplementary material 1 (pdf 152 KB)


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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Yingzhen Yang
    • 1
    Email author
  • Jiashi Feng
    • 2
  • Nebojsa Jojic
    • 3
  • Jianchao Yang
    • 4
  • Thomas S. Huang
    • 1
  1. 1.Beckman Institute, University of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of ECENational University of SingaporeSingaporeSingapore
  3. 3.Microsoft ResearchRedmondUSA
  4. 4.SnapchatLos AngelesUSA

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