UC Davis

We analyze geometric aspects of clustering procedures based on low-dimensional embeddings. In particular, we are interested in understanding the occurrence of the so-called orthogonal cone structure (OCS) that can be observed empirically in various low-dimensional embeddings, including kernel PCA, spectral clustering, Isomap, and clustering based on the Hodge Laplacian. Inspired by recent work on the OCS based on graph Laplacians, we study OCS in the context of weighted Laplacian and kernel PCA. This involves the development of a notion of a well-separated mixture model and other characteristics of the methodology. These characteristics are then used to quantify the OCS. We illustrate this for weighted Laplacian and kernel PCA in both the population setting and the sample setting.