On the LASSO and Dantzig selector equivalence

Recovery of sparse signals from noisy observations is a problem that arises in many information processing contexts. LASSO and the Dantzig selector (DS) are two well-known schemes used to recover high-dimensional sparse signals from linear observations. This paper presents some results on the equivalence between LASSO and DS. We discuss a set of conditions under which the solutions of LASSO and DS are same. With these conditions in place, we formulate a shrinkage procedure for which LASSO and DS follow the same solution path. Furthermore, we show that under these shrinkage conditions the solution to LASSO and DS can be attained in at most S homotopy steps, where S is the number of nonzero elements in the final solution. Thus the computational cost for finding complete homotopy path for an M × N system is merely O(SMN).