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SpectralNet and Ncut Loss

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- 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. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Ì‡SN I 1 Spectral Clustering & Ncut 2 Normalized Cut Loss for Weakly-supervised CNN Segmentation 3 SPECTRALNET ×n Ncut loss & SpectralNet
- 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. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET ×n Ncut loss & SpectralNet
- 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. 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. 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. 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. 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. 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. 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. 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. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Algorithm ×n Ncut loss & SpectralNet
- 14. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET Results ×n Ncut loss & SpectralNet

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