Algorithms of fuzzy clustering with supervised learning

In [CORE Discussion Paper 2018/1], Y. Nesterov suggests a new model for fuzzy clustering by minimizing a strongly convex potential function. This new technique based on an electoral model has the two main advantages that under some natural assumptions the problem has a unique fixed point and that there exist polynomial-time algorithms to solve this problem. The solution can be either found by an alternating minimization scheme simulating sequential elections or by direct minimization of the strongly convex potential function. Therefore, the first part of this thesis is dedicated to the implementation and the numerical analysis of the different algorithms to solve the new suggested model. Nevertheless, the particularity of the new proposed model, referenced in this document under the name "Simplified Soft Clustering", is to require prior knowledge about the centers of the clusters. This particularity makes the use of the new technique complicated for unsupervised learning. The goal of the second part of this document is therefore to compare the new clustering strategy with other classical machine learning algorithms, aiming to answer the question "For what can we use the simplified soft clustering model ?".