On LARS/homotopy equivalence conditions for over-determined LASSO
Résumé
We revisit the positive cone condition given by Efron et al. [1] 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) [1] and homotopy [2] 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 [3], ensuring the exact recovery of any k-sparse representation using both LARS and homotopy.