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Subspace Learning by \(\ell ^{0}\)-Induced Sparsity

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Abstract

Subspace clustering methods partition the data that lie in or close to a union of subspaces in accordance with the subspace structure. Such methods with sparsity prior, such as sparse subspace clustering (SSC) (Elhamifar and Vidal in IEEE Trans Pattern Anal Mach Intell 35(11):2765–2781, 2013) with the sparsity induced by the \(\ell ^{1}\)-norm, are demonstrated to be effective in subspace clustering. Most of those methods require certain assumptions, e.g. independence or disjointness, on the subspaces. However, 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 \(\ell ^{0}\)-SSC guarantees the subspace-sparse representation, a key element in subspace clustering, 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 which shows that obtaining the subspace representation under our general assumptions can not be much computationally cheaper than solving the corresponding \(\ell ^{0}\) sparse representation problem of \(\ell ^{0}\)-SSC. A novel approximate algorithm named Approximate \(\ell ^{0}\)-SSC (A\(\ell ^{0}\)-SSC) is developed which employs proximal gradient descent to obtain a sub-optimal solution to the optimization problem of \(\ell ^{0}\)-SSC with theoretical guarantee. The sub-optimal solution is used to build a sparse similarity matrix upon which spectral clustering is performed for the final clustering results. Extensive experimental results on various data sets demonstrate the superiority of A\(\ell ^{0}\)-SSC compared to other competing clustering methods. Furthermore, we extend \(\ell ^{0}\)-SSC to semi-supervised learning by performing label propagation on the sparse similarity matrix learnt by A\(\ell ^{0}\)-SSC and demonstrate the effectiveness of the resultant semi-supervised learning method termed \(\ell ^{0}\)-sparse subspace label propagation (\(\ell ^{0}\)-SSLP).

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Notes

  1. Continuous distribution here indicates that the data distribution is non-degenerate in the sense that the probability measure of any hyperplane of dimension less than that of the subspace is 0.

  2. Even one would stick to the very original formulation without noise tolerance, (4) is still equivalent to (7) with some Lagrangian multiplier \(\lambda \).

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Authors and Affiliations

  1. Snap Research, Venice, USA

    Yingzhen Yang & Jianchao Yang

  2. Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, USA

    Yingzhen Yang & Thomas S. Huang

  3. Department of ECE, National University of Singapore, Singapore, Singapore

    Jiashi Feng

  4. Microsoft Research, Redmond, USA

    Nebojsa Jojic

Authors
  1. Yingzhen Yang
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  2. Jiashi Feng
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  3. Nebojsa Jojic
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  4. Jianchao Yang
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  5. Thomas S. Huang
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Correspondence to Yingzhen Yang.

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Communicated by Edwin Hancock, Richard Wilson, Will Smith, Adrian Bors, Nick Pears.

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This material is based upon work supported by the findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The work of Yingzhen Yang was supported in part by an IBM gift grant to Beckman Institute, UIUC. The work of Jiashi Feng was partially supported by National University of Singapore startup Grant R-263-000-C08-133 and Ministry of Education of Singapore AcRF Tier One Grant R-263-000-C21-112.

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Yang, Y., Feng, J., Jojic, N. et al. Subspace Learning by \(\ell ^{0}\)-Induced Sparsity. Int J Comput Vis 126, 1138–1156 (2018). https://doi.org/10.1007/s11263-018-1092-4

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