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
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