The momentum term has long been used in machine learning algorithms, especially back-propagation, to improve their speed of convergence. In this paper, we derive an expression to prove the O ( 1 / k 2 ) convergence rate of the online gradient method, with momentum type updates, when the individual gradients are constrained by a growth condition. We then apply these type of updates to video background modelling by using it in the update equations of the Region-based Mixture of Gaussians algorithm. Extensive evaluations are performed on both simulated data, as well as challenging real world scenarios with dynamic backgrounds, to show that these regularised updates help the mixtures converge faster than the conventional approach and consequently improve the algorithm’s performance. 
Fast convergence of regularised Region-based Mixture of Gaussians for dynamic background modelling