Piecewise Linearity of Min-Norm Solution Map of a Nonconvexly Regularized Convex Sparse Model

It is well known that the minimum ℓ2-norm solution of the convex LASSO model, say xstar, is a continuous piecewise linear function of the regularization parameter λ, and its signed sparsity pattern is constant within each linear piece. The current study is an extension of this classic result, proving that the aforementioned properties extend to the min-norm solution map xstar(y,λ), where y is the observed signal, for a generalization of LASSO termed the scaled generalized minimax concave (sGMC) model. The sGMC model adopts a nonconvex debiased variant of the ℓ1-norm as sparse regularizer, but its objective function is overall-convex. Based on the geometric properties of xstar(y,λ), we propose an extension of the least angle regression (LARS) algorithm, which iteratively computes the closed-form expression of xstar(y,λ) in each linear zone. Under suitable conditions, the proposed algorithm provably obtains the whole solution map xstar(y,λ) within finite iterations. Notably, our proof techniques for establishing continuity and piecewise linearity of xstar(y,λ) are novel, and they lead to two side contributions: (a) our proofs establish continuity of the sGMC solution set as a set-valued mapping of (y,λ); (b) to prove piecewise linearity and piecewise constant sparsity pattern of xstar(y,λ), we do not require any assumption that previous work relies on (whereas to prove some additional properties of xstar(y,λ), we use a different set of assumptions from previous work).