A sparser reduced set density estimator by introducing weighted l₁ penalty term

Reduced set density estimator (RSDE), employing a small percentage of available data samples, is an effective and important nonparametric technique for probability density function estimation. Despite that RSDE has been demonstrated to perform better in the computational complexity compared with several existing approaches, it still faces the critical challenge in practical applications when training the estimator on large data sets. Dealing with its high complexity both in time and space, a sparser reduced set density estimator with weighted l₁ penalty term (RSDE-WL₁) is proposed in this paper. By introducing the weighted l₁ norm of the estimated coefficients as additional penalty term, the sparse weights of density estimation are estimated, in which small weight coefficients are more likely to be driven to zero. The proposed iterative algorithm is used to solve the corresponding convex optimization problem efficiently. Some discussions about the choice of parameters and the trade-off between sparsity and accuracy are also given. The simulations of several examples demonstrate that the proposed RSDE-WL₁ is superior to the related methods including the RSDE in sparsity and complexity. While requiring less number of kernels in density estimation, our approach is comparable to the full-sample optimized Parzen window estimator in terms of test accuracy.