Convex Approximation to the Integral Mixture Models Using Step Functions

The parameter estimation to mixture models has been shown as a local optimal solution for decades. In this paper, we propose a functional estimation to mixture models using step functions. We show that the proposed functional inference yields a convex formulation and consequently the mixture models are feasible for a global optimum inference. The proposed approach further unifies the existing isolated exemplar-based clustering techniques at a higher level of generality, e.g. it provides a theoretical justification for the heuristics of the clustering by affinity propagation Frey & Dueck (2007), it reproduces Lashkari & Golland (2007)'s's convex formulation as a special case under this step function construction. Empirical studies also verify the theoretic justifications.