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.

Quantitative normal approximation bounds are important to obtain finite-sample, non-asymptotic inferential guarantees for various statistical problems. We derived such quantitative normal approximation results for a class of stabilizing statistics. Examples of such statistics include $k$-nearest neighbor based Entropy estimators, Euler characteristics, minimal spanning trees and Random forests. The central challenge in these problems is to handle the delicate dependency structure arising in such statistics. We handle this by utilizing the concept of stabilization, which we combine with Stein's method to establish our results. This notion of stabilization is quite universal in characterizing the local dependencies and thus provides a powerful tool for obtaining normal approximation analysis and doing finite-sample statistical inferences. Additionally, bootstrap methodology can be considered for performing practical statistical inference for the above class of problems.