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Spectral Clustering
Spectral Clustering
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Spectral clustering constructs a similarity graph and computes eigenvectors of the graph Laplacian
L
=
D
−
W
(or its normalized variant). Why does the multiplicity of the zero eigenvalue of
L
equal the number of connected components in the graph?
Hide and think first
A.
The Laplacian is always rank-deficient with exactly half of its eigenvalues equal to zero, regardless of connectivity
B.
The zero eigenvalue of the Laplacian always has multiplicity exactly one for every graph, regardless of connectivity
C.
Each component's indicator vector is an independent null vector of
L
, and no cross-component combination adds another
D.
The result only holds for unweighted graphs with binary adjacency, not for general weighted similarity graphs
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