Adaptive filtering algorithms for promoting sparsity

We provide a mathematical framework for developing adaptive filtering algorithms for exploiting/enforcing sparsity. The approach is based on minimizing a regularized mean squared error criterion with sparsity being promoted by the regularizing term which consists of a diversity measure. A steepest descent algorithm (SDA) is developed to minimize the regularized cost function. Then we extend the algorithm to the adaptive environment and develop a class of algorithms, which we term the pLMS algorithm class and which incudes important variants - pLLMS (leaky pLMS) and pNLMS (normalized pLMS). The framework is quite general and encompasses a broad range of adaptive algorithms with the pNLMS having similarity with the proportionate normalized least-mean-squares (PNLMS) algorithm. Computer simulations have been conducted using the echo canceller application as an example of a sparse environment. The simulations clearly show the ability of the developed algorithms to exploit the inherent sparsity structure, thereby outperforming conventional algorithms like the NLMS algorithm in this application.