A Batch Rival Penalized Expectation-Maximization Algorithm for Gaussian Mixture Clustering with Automatic Model Selection

Algorithm 1

Adaptive RPEM algorithm.

Initialization: Given a specific 𝑘 (𝑘≥𝑘^∗, 𝑘^∗ is the true number of clusters),

       initialize the parameter Θ.

Step  1: Given the current input 𝐱_𝑡 and the parameter estimate, written

   as Θ^(𝑛), compute ℎ(𝑗∣𝐱_𝑡,Θ^(𝑛))’s and 𝑔(𝑗∣𝐱_𝑡,Θ^(𝑛))’s via(2)

   and (7), respectively.

Step  2: Given ℎ(𝑗∣𝐱_𝑡,Θ^(𝑛))’s and 𝑔(𝑗∣𝐱_𝑡,Θ^(𝑛))’s, we update Θ by

Θ^(𝑛+1)=Θ^(𝑛)+𝜂(𝜔_𝑡(Θ;𝐱_𝑡)/Θ)∣_Θ(𝑛),

   with

𝜔_𝑡(Θ;𝐱_𝑡)=∑^𝑘_𝑗=1𝑔(𝑗𝐱_𝑡,Θ)ln[𝛼_𝑗𝑝(𝐱_𝑡𝜽_𝑗)],

   where 𝜂 is a small positive learning rate.

Step  3: Let𝑛=𝑛+1, and go to Step  1 for the next iteration until Θ

       is converged.