Abstract
The paper investigates several notions of graph Laplacians and graph kernels from the perspective of understanding the graph clustering via the graph embedding into an Euclidean space. We propose hereby a unified view of spectral graph clustering and kernel clustering methods. The various embedding techniques are evaluated from the point of view of clustering stability (with respect to k-means that is the algorithm underpinning the spectral and kernel methods). It is shown that the choice of a fixed number of dimensions may result in clustering instability due to eigenvalue ties. Furthermore, it is shown that kernel methods are less sensitive to the number of used dimensions due to downgrading the impact of less discriminative dimensions.
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Notes
- 1.
The graphs may originate from data embedded in Euclidean space, for which a graph was constructed based on thresholding similarities/distances between the objects [12].
- 2.
\(s_{ii}=0\) is set for technical reasons. Self-similarity does not affect the basic formula, but the value 0 is useful when handling Laplacians.
- 3.
As the matrix S is symmetric, this definition is a bit too laborious, but later on we will refer to asymmetric matrices.
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Wierzchoń, S.T., Kłopotek, M.A. (2020). Spectral Cluster Maps Versus Spectral Clustering. In: Saeed, K., Dvorský, J. (eds) Computer Information Systems and Industrial Management. CISIM 2020. Lecture Notes in Computer Science(), vol 12133. Springer, Cham. https://doi.org/10.1007/978-3-030-47679-3_40
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