Optimum “Eye Location” Problem for Spectral Clustering With Cosine Distance

Abstract

It has recently been reported that Spectral Clustering gives state-of-the art clustering performance for many real-life benchmark datasets. When building the dissimilarity (distance) matrix for the Laplacian matrix, cosine distance is also reported to give the best performance among the other distance types for various real-life datasets. In this paper, we introduce an Optimum Sphere Location Problem for Spectral Clustering with cosine distance, and our investigations result in novel results, some of which are as follows: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> ) The Optimum “Eye Location” Problem (a.k.a. Optimum Sphere Location Problem) is a challenging problem due to the high complexity between the center of the unit sphere and the eigenvalues of the corresponding Laplacian matrix. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ii</i> ) As compared to the sphere at the origin, choosing the “eye location” at the data mean generally increases the eigenvalues of the Laplacian matrix. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iii</i> ) As compared to the sphere at the data mean, choosing the “eye location” at an appropriate point on the 2nd or 3rd dominant eigenvector of the data covariance matrix potentially gives better clustering performance for the same spectral clustering algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iv</i> ) The proposed method includes the traditional methods, which are locating the unit sphere at the origin and at the data mean, as special cases. Computer simulations confirm the findings.