Hierarchical clustering algorithms, including agglomerative linkage-based methods, have been designed without an objective function in mind. We present the work of Addad et al., who build on the work of Dasgupta (2016) in defining an appropriate cost function which should be optimized by the hierarchy a clustering produces. The authors define "admissible" cost functions as those which find their optimum when a clustering recovers a "ground truth" from which an input was generated, and show that these functions have certain properties. Under a suitable cost function, the authors analyze the performance of linkage-based and other clustering algorithms. They present a new algorithm which improves on the worst-case performance of these algorithms.
Based on a paper by Vincent Cohen-Addad, Varun Kanade, Frederik Mallman-Trenn, Claire Mathieu: https://epubs.siam.org/doi/10.1137/1.9781611975031.26