Transductive-Inductive Cluster Approximation Via Multivariate Chebyshev Inequality

TL;DR

This paper introduces a transductive-inductive clustering algorithm using multivariate Chebyshev inequality, providing theoretical guarantees on error bounds and demonstrating stability in estimating the number of clusters for multidimensional data.

Contribution

It formulates a novel clustering method based on Chebyshev inequality with proven convergence and error bounds, applicable across various multidimensional datasets.

Findings

Theoretical proof of convergence of reconstruction error.

Error bounds for clustering accuracy.

Empirical stability in estimating the number of clusters.

Abstract

Approximating adequate number of clusters in multidimensional data is an open area of research, given a level of compromise made on the quality of acceptable results. The manuscript addresses the issue by formulating a transductive inductive learning algorithm which uses multivariate Chebyshev inequality. Considering clustering problem in imaging, theoretical proofs for a particular level of compromise are derived to show the convergence of the reconstruction error to a finite value with increasing (a) number of unseen examples and (b) the number of clusters, respectively. Upper bounds for these error rates are also proved. Non-parametric estimates of these error from a random sample of sequences empirically point to a stable number of clusters. Lastly, the generalization of algorithm can be applied to multidimensional data sets from different fields.