Quantifying uncertainty in spectral clusterings: expectations for perturbed and incomplete data

TL;DR

This work discusses a mathematical framework based on random set theory for the computational Monte Carlo approximation of statistically expected clusterings in case of corrupted data, and proposes several computationally accessible quantities of interest.

Abstract

Spectral clustering is a popular unsupervised learning technique which is able to partition unlabelled data into disjoint clusters of distinct shapes. However, the data under consideration are often experimental data, implying that the data is subject to measurement errors and measurements may even be lost or invalid. These uncertainties in the corrupted input data induce corresponding uncertainties in the resulting clusters, and the clusterings thus become unreliable. Modelling the uncertainties as random processes, we discuss a mathematical framework based on random set theory for the computational Monte Carlo approximation of statistically expected clusterings in case of corrupted, i.e., perturbed, incomplete, and possibly even additional, data. We propose several computationally accessible quantities of interest and analyze their consistency in the infinite data point and infinite Monte Carlo sample limit. Numerical experiments are provided to illustrate and compare the proposed quantities.

Figures (11)

• Figure 1: \newlabelfig:pi0 Illustration of the reference data set $X$ (left), with the orange data points vanishing after perturbation, and the corrupted data set $\pi(\omega)=\pi_1(\omega)\cup\pi_2(\omega)$ (right). Here $\pi_1(\omega)$ are the perturbed, non-vanishing points from $X$ (blue) and $\pi_2(\omega)$ is a set of random additional data (green). The illustration of the corrupted data set on the right shows the reference data set in transparent colors.
• Figure 1: Sampled reference data sets of the point cloud in circle data set for $n=400$ (left) and $n=1600$ (right).
• Figure 2: Expected misclustering rate for the point cloud in circle data set for $n=400$ (left) and $n=1600$ (right).
• Figure 3: Approximated coverage function, Vorob'ev expectation, ODF-expectation, and spectral expectation for different noise levels for the point in circle data set with $n=400$.
• Figure 4: Approximated coverage function, Vorob'ev expectation, ODF-expectation, and spectral expectation for different noise levels for the point in circle data set with $n=1600$.

Theorems & Definitions (11)

• Theorem 1: VBB2008
• Definition 1
• Theorem 1: VBB2008
• Theorem 2: VBB2008
• Theorem 3: Strong law of large numbers
• Theorem 4: CGR2006
• Corollary 5
• Proof 1
• Theorem 6
• Remark 1