Graph based Subspace Segmentation Canyi Lu National University of - - PowerPoint PPT Presentation

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Graph based Subspace Segmentation Canyi Lu National University of - - PowerPoint PPT Presentation

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Graph based Subspace Segmentation Canyi Lu National University of Singapore Nov. 21, 2013 Content Subspace Segmentation Problem Related Work Sparse Subspace Clustering (SSC) Low-Rank Representation (LRR) Multi-Subspace

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Graph based Subspace Segmentation

Canyi Lu National University of Singapore

  • Nov. 21, 2013
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Content

  • Subspace Segmentation Problem
  • Related Work
  • Sparse Subspace Clustering (SSC)
  • Low-Rank Representation (LRR)
  • Multi-Subspace Representation (MSR)
  • Subspace Segmentation via Quadratic Programming (SSQP)
  • Least Squares Regression (LSR)
  • Enforced Block Diagonal Conditions
  • Correlation Adaptive Subspace Segmentation (CASS)
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Subspace Segmentation

  • Given sufficient data points drawn from multiple subspaces,

the goal is to find

  • the number of subspaces
  • their dimensions
  • a basis of each subspace
  • the segmentation of the data corresponding to different

subspaces

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Applications

  • Face clustering
  • Motion segmentation
  • Image segmentation
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Algorithm

Spectral Clustering：

  • Graph construction: construct a graph (affinity matrix) to

measure the similarities between data points

  • Segment the data points into multiple clusters
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Subspace Assumption

  • Disjoint subspaces
  • Independent subspaces
  • Orthogonal subspaces
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Previous Work: SSC

  • Graph construction by sparse representation

Bin Cheng, Jianchao Yang, Shuicheng Yan, Yun Fu, Thomas S. Huang, Learning with l1-graph for image analysis. TIP, 2010 Elhamifar, E. and R. Vidal. Sparse Subspace Clustering. CVPR 2009

  • The solution to the above L1 minimization problem is

block diagonal when the data are from independent subspace.

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Previous Work: LRR

  • Graph construction by low rank representation

Liu, G., Z. Lin, S. Yan, J. Sun, Y. Yu and Y. Ma. Robust recovery of subspace structures by low-rank representation. TPAMI. 2013

  • The solution to the above nuclear norm minimization

problem is block diagonal when the data are from independent subspace.

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Previous Work: MSR

  • Graph construction by low rank representation
  • D. Luo, et al., “Multi-subspace representation and discovery,” in ECML, 2011
  • The solution to the above minimization problem is block

diagonal when the data are from independent subspace.

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Previous Work: SSQP

  • Subspace Segmentation via Quadratic Programming

Shusen Wang, Xiaotong Yuan, Tiansheng Yao, Shuicheng Yan, Jialie Shen. Efficient Subspace Segmentation via Quadratic Programming. AAAI. 2011.

  • The solution to the above nuclear norm minimization is

block diagonal when the data are from orthogonal subspace.

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Least Squares Regression

  • Subspace Segmentation via Least Squares Regression

Canyi Lu, Hai Min, Shuicheng Yan. Efficient Subspace Segmentation via Least Squares

  • Regression. ECCV. 2012.
  • The solution to the above minimization is block diagonal

when the data are from independent subspace.

  • Grouping effect of LSR (in vector form)
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  • Consider the following general problem
  • What kind of objective function involves the block

diagonal property under certain condition?

Canyi Lu, Hai Min, Shuicheng Yan. Efficient Subspace Segmentation via Least Squares

  • Regression. ECCV. 2012.
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Enforced Block Diagonal Conditions

Canyi Lu, Hai Min, Shuicheng Yan. Efficient Subspace Segmentation via Least Squares

  • Regression. ECCV. 2012.
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Block Diagonal Property: independent subspaces

Canyi Lu, Hai Min, Shuicheng Yan. Efficient Subspace Segmentation via Least Squares

  • Regression. ECCV. 2012.
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Block Diagonal Property

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Block Diagonal Property : orthogonal subspaces

Block diagonal property when the subspaces are orthogonal

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Block Diagonal Property : disjoint subspaces

  • Block diagonal property by SSC on disjoint subspaces

Elhamifar, E. and R. Vidal. Clustering disjoint subspaces via sparse representation. ICASS, 2010

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Block Diagonal Property : disjoint subspaces

  • Block diagonal property by LSR on disjoint subspaces
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Correlation Adaptive Subspace Segmentation

  • A better choice: balance the sparsity and grouping effect
  • Correlation Adaptive Subspace Segmentation (CASS)
  • C. Lu, et al. Correlation Adaptive Subspace Segmentation by Trace Lasso.
  • ICCV. 2013.
  • If the data are uncorrelated (the data points are
  • rthogonal )
  • If the data are highly correlated (the data points are all

the same, )

  • For other case,
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Correlation Adaptive Subspace Segmentation

  • C. Lu, et al. Correlation Adaptive Subspace Segmentation by Trace Lasso.
  • ICCV. 2013.
  • CASS also leads to block sparse solution when the data

are from independent subspace.

  • Grouping effect of CASS
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Correlation Adaptive Subspace Segmentation

Comparison of different affinity matrices

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Experiments: Motion Segmentation

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Experiments: Face Clustering

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Thanks!