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Clustering and Segmentation

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Discrete Calculus

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Abstract

Clustering algorithms are used to find communities of nodes that all belong to the same group. This grouping process is also known as image segmentation in image processing. The clustering problem is also deeply connected to machine learning because a solution to the clustering problem may be used to propagate labels from observed data to unobserved data. In general network analysis, the identification of a grouping allows for the analysis of the nodes within each group as separate entities. In this chapter, we use the tools of discrete calculus to examine both the targeted clustering problem (i.e., finding a specific group) and the untargeted clustering problem (i.e., discovering all groups). We additionally show how to apply these clustering models to the clustering of higher-order cells, e.g., to cluster edges.

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Notes

  1. 1.

    In the rest of this chapter we treated the data as univariate in order to simplify the exposition, with the understanding that all of the machinery could also be applied to multivariate data. However, since k-means is almost exclusively applied to multivariate data we have adopted a multivariate view of data in this section. Therefore, it is assumed that each node (data point) is associated with a tuple of data, rather than a scalar.

  2. 2.

    Some authors have tried to incorporate spatial location into k-means by using the pixel coordinates as part of the feature vector in the application of k-means. This device can mitigate the problem described here in certain circumstances, but does not generalize to applications in which the network has no embedding or when the embedding is complicated, as in the gene expression example in Sect. 6.5.4 or the geospatial example in Sect. 5.9.4.

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Authors and Affiliations

  1. Siemens Corporate Research, 755 College Road East, Princeton, NJ, 08540-6632, USA

    Dr. Leo J. Grady

  2. Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, 02129, USA

    Dr. Jonathan R. Polimeni

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Grady, L.J., Polimeni, J.R. (2010). Clustering and Segmentation. In: Discrete Calculus. Springer, London. https://doi.org/10.1007/978-1-84996-290-2_6

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