In this paper, a novel way of deriving proportionate adaptive filters is proposed based on diversity measure minimization using the iterative reweighting techniques well-known in the sparse signal recovery (SSR) area. The resulting least mean square (LMS)-type and normalized LMS (NLMS)-type sparse adaptive filtering algorithms can incorporate various diversity measures that have proved effective in SSR. Furthermore, by setting the regularization coefficient of the diversity measure term to zero in the resulting algorithms, Sparsity promoting LMS (SLMS) and Sparsity promoting NLMS (SNLMS) are introduced, which exploit but do not strictly enforce the sparsity of the system response if it already exists. Moreover, unlike most existing proportionate algorithms that design the step-size control factors based on heuristics, our SSR-based framework leads to designing the factors in a more systematic way. Simulation results are presented to demonstrate the convergence behavior of the derived algorithms for systems with different sparsity levels.