On LARS/Homotopy Equivalence Conditions for Over-Determined LASSO

We revisit the positive cone condition given by Efron for the over-determined least absolute shrinkage and selection operator (LASSO). It is a sufficient condition ensuring that the number of nonzero entries in the solution vector keeps increasing when the penalty parameter decreases, based on which the least angle regression (LARS) and homotopy algorithms yield the same iterates. We show that the positive cone condition is equivalent to the diagonal dominance of the Gram matrix inverse, leading to a simpler way to check the positive cone condition in practice. Moreover, we elaborate on a connection between the positive cone condition and the mutual coherence condition given by Donoho and Tsaig , ensuring the exact recovery of any k -sparse representation using both LARS and homotopy.