Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

SpectralNet and Ncut Loss

84 views

Published on

SpectralNet and Ncut Loss

Published in: Data & Analytics
  • Be the first to comment

  • Be the first to like this

SpectralNet and Ncut Loss

  1. 1. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Ncut loss & SpectralNet ×n UTS Building 11 2018 c4 14F ×n Ncut loss & SpectralNet
  2. 2. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET ̇SN I 1 Spectral Clustering & Ncut 2 Normalized Cut Loss for Weakly-supervised CNN Segmentation 3 SPECTRALNET ×n Ncut loss & SpectralNet
  3. 3. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Outline 1 Spectral Clustering & Ncut 2 Normalized Cut Loss for Weakly-supervised CNN Segmentation 3 SPECTRALNET ×n Ncut loss & SpectralNet
  4. 4. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET ×n Ncut loss & SpectralNet
  5. 5. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET The objective function of Spectral Clustering is given by, min H Tr(H LH) s.t. H ∈ Rn×k , H H = I where H is the low dimensional embedding matrix, Tr(·) is the trace operator, L = I − D− 1 2 WD− 1 2 , D is a diagonal matrix with each diagonal element djj = n i=1 wij, and I is an identity matrix. Theorem The number k of connected components of the graph is equal to the multiplicity of 0 as an eigenvalue of L . Proof. Since L is positive semi-definite, all eigenvalues of L are non-negative. It is straightforward to check that 1 2 n i,j=1 wij hi − hj 2 = 0 while wij ≥ 0, if and only if h is constant on each connected component. ×n Ncut loss & SpectralNet
  6. 6. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Outline 1 Spectral Clustering & Ncut 2 Normalized Cut Loss for Weakly-supervised CNN Segmentation 3 SPECTRALNET ×n Ncut loss & SpectralNet
  7. 7. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET           1 1 1 1 0 0 0                     0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 0 0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0.5 0 0 0 0 0.5 0 0.5 0 0 0 0 0.5 0.5 0                     0 0 0 0 1 1 1           = 0 ×n Ncut loss & SpectralNet
  8. 8. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET           1 1 1 1 1 0 0                     0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 0 0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0.5 0 0 0 0 0.5 0 0.5 0 0 0 0 0.5 0.5 0                     0 0 0 0 0 1 1           = 1 ×n Ncut loss & SpectralNet
  9. 9. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Ncut loss = k Sk W(1 − Sk) d Sk ×n Ncut loss & SpectralNet
  10. 10. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Outline 1 Spectral Clustering & Ncut 2 Normalized Cut Loss for Weakly-supervised CNN Segmentation 3 SPECTRALNET ×n Ncut loss & SpectralNet
  11. 11. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Two Limitations of Spectral Clustering: Scalability Generalization of the Spectral Embedding, i.e., out-of-sample extension. ×n Ncut loss & SpectralNet
  12. 12. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET hi = fθ(xi) is a general neural network, and its lose function is, loss = 1 m2 m i,j=1 wij hi − hj 2 2 Suppose the last layer of network is ˜H ∈ Rk×m. We need to constrain H H = I. A linear map that orthogonalizes the columns of ˜H is computed through its QR decomposition ˜H = QL . L is obtained by Cholesky decomposition ˜H ˜H, ˜H ˜H = LL , Then, Q Q = L−1 ˜H ˜H(L−1 ) = L−1 LL (L−1 ) = (L−1 L) = I Thus, Q = ˜H(L−1) © ×n Ncut loss & SpectralNet
  13. 13. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Algorithm ×n Ncut loss & SpectralNet
  14. 14. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Results ×n Ncut loss & SpectralNet

×