In this paper we introduce a method for person re-identification based on discriminative, sparse basis expansions of targets in terms of a labeled gallery of known individuals. We propose an iterative extension to sparse discriminative classifiers capable of ranking many candidate targets. The approach makes use of soft- and hard- re-weighting to redistribute energy among the most relevant contributing elements and to ensure that the best candidates are ranked at each iteration. Our approach also leverages a novel visual descriptor which we show to be discriminative while remaining robust to pose and illumination variations. An extensive comparative evaluation is given demonstrating that our approach achieves state-of-the-art performance on single- and multi-shot person re-identification scenarios on the VIPeR, i-LIDS, ETHZ, and CAVIAR4REID datasets. The combination of our descriptor and iterative sparse basis expansion improves state-of-the-art rank-1 performance by six percentage points on VIPeR and by 20 on CAVIAR4REID compared to other methods with a single gallery image per person. With multiple gallery and probe images per person our approach improves by 17 percentage points the state-of-the-art on i-LIDS and by 72 on CAVIAR4REID at rank-1. The approach is also quite efficient, capable of single-shot person re-identification over galleries containing hundreds of individuals at about 30 re-identifications per second.