Spectral graph clustering is among the most popular algorithms for unsupervised segmentation. Applications include problems such as speech separation, segmenting motions or objects in video sequences and community detection in social media. It is based on the computation of a few eigenvectors of a matrix defining the connections between the graph nodes.
In many real world applications, not all edge weights can be defined. In video sequences, for instance, not all 3d-points of the observed objects are visible in all the images. Relations between graph nodes representing the 3d-points cannot be defined if these never co-occur in the same images. It is common practice to simply assign an affinity of zero to such edges.
In this article, we present a formal proof that this procedure decreases the separation between two clusters. An upper bound is derived on the second smallest eigenvalue of the Laplacian matrix. Furthermore, an algorithm to infer missing edges is proposed and results on synthetic and real image data are presented.