In this paper, we introduce two indoor Wireless Local Area Network (WLAN) positioning methods using augmented sparse recovery algorithms. These schemes render a sparse user's position vector, and in parallel, minimize the distance between the online measurement and radio map. The overall localization scheme for both methods consists of three steps: 1) coarse localization, obtained from comparing the online measurements with clustered radio map. A novel graph-based method is proposed to cluster the offline fingerprints. In the online phase, a Region Of Interest (ROI) is selected within which we search for the user's location; 2) Access Point (AP) selection; and 3) fine localization through the novel sparse recovery algorithms. Since the online measurements are subject to inordinate measurement readings, called outliers, the sparse recovery methods are modified in order to jointly estimate the outliers and user's position vector. The outlier detection procedure identifies the APs whose readings are either not available or erroneous. The proposed localization methods have been tested with Received Signal Strength (RSS) measurements in a typical office environment and the results show that they can localize the user with significantly high accuracy and resolution which is superior to the results from competing WLAN fingerprinting localization methods.