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We present a proximal quasi-Newton method in which the approximation of theHessian has the special format of "identity minus rank one" (IMRO) in eachiteration. The proposed structure enables us to effectively recover theproximal point. The algorithm is applied to $l_1$-regularized least squareproblem arising in many applications including sparse recovery in compressivesensing, machine learning and statistics. Our numerical experiment suggeststhat the proposed technique competes favourably with other state-of-the-artsolvers for this class of problems. We also provide a complexity analysis forvariants of IMRO, showing that it matches known best bounds.

IMRO: a proximal quasi-Newton method for solving $l_1$-regularized least square problem