Can Critical-Point Paths Under ${\ell}_{{p}}$ -Regularization $(0<p<1)$ Reach the Sparsest Least Squares Solutions?

Abstract

The solution path of the least square problem under ℓ_p-regularization (0 <; p <; 1) is studied, where the Lagrangian multiplier λ due to the constraint is the parameter of the path. It is first proven that the least square solution of an unconstrained overdetermined linear system is connected with the origin, under a mild condition, by a continuous path of critical points of an ℓ_p-regularized squared error function. Based on this fact, it is proven that every sparsest least square solution of an underdetermined system is connected with the origin by a critical-point path. The existence theorem holds more generally for any least square solution whose support has its associated submatrix of the fat sensing matrix be full column rank. This is a sufficient condition for the existence, and allows to reduce the underdetermined problem to an overdetermined one with the off-support variable(s) nullified. A necessary condition is that the gradient of the ℓ_p regularizer with respect to the support variables lies in the row space of the submatrix (which is not necessarily full column rank).