Soft clustering by convex electoral model

In this paper, we suggest a new technique for soft clustering of multidimensional data. It is based on a new convex voting model, where each voter chooses a party with certain probability depending on the divergence between his/her preferences and the position of the party. The parties can react on the results of polls by changing their positions. We prove that under some natural assumptions this system has a unique fixed point, providing a unique solution for soft clustering. The solution of our model can be found either by imitation of the sequential elections, or by direct minimization of a convex potential function. In both cases, the methods converge linearly to the solution. We provide our methods with worst-case complexity bounds. To the best of our knowledge, these are the first polynomial-time complexity results in this field.