Abstract L1 -regularized least squares, with the ability of discovering sparse representations, is quite prevalent in the ?eld of machine learning, statistics and signal processing. In this paper, we propose a novel algorithm called Dual Projected Newton Method (DPNM) to solve the 1 -regularized least squares problem. In DPNM, we ?rst derive a new dual problem as a box constrained quadratic programming. Then, a projected Newton method is utilized to solve the dual problem, achieving a quadratic convergence rate. Moreover, we propose to utilize some practical techniques, thus it greatly reduces the computational cost and makes DPNM more ef?cient. Experimental results on six real-world data sets indicate that DPNM is very ef?cient for solving the 1 -regularized least squares problem, by comparing it with state of the art methods.