SpectralNet and Ncut Loss

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
̇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
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×n Ncut loss & SpectralNet

8. Spectral Clustering & Ncut Normalized Cut Loss for Weakly-supervised CNN Segmentation SPECTRALNET
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×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