# Semi-supervised Clustering with Two Types of Background Knowledge: Fusing Pairwise Constraints and Monotonicity Constraints

## TL;DR

This paper introduces a novel semi-supervised clustering method that integrates pairwise constraints and monotonicity constraints using a new distance measure and EM optimization, effectively fusing two types of background knowledge.

## Contribution

It is the first method to combine pairwise and monotonicity constraints in clustering, providing a formal framework and an optimization scheme for this integration.

## Key findings

- Effective in benchmark datasets
- Successful application to real-world data
- Outperforms existing clustering methods

## Abstract

This study addresses the problem of performing clustering in the presence of two types of background knowledge: pairwise constraints and monotonicity constraints. To achieve this, the formal framework to perform clustering under monotonicity constraints is, firstly, defined, resulting in a specific distance measure. Pairwise constraints are integrated afterwards by designing an objective function which combines the proposed distance measure and a pairwise constraint-based penalty term, in order to fuse both types of information. This objective function can be optimized with an EM optimization scheme. The proposed method serves as the first approach to the problem it addresses, as it is the first method designed to work with the two types of background knowledge mentioned above. Our proposal is tested in a variety of benchmark datasets and in a real-world case of study.