Piecewise Linearity of Min-Norm Solution Map of a Nonconvexly Regularized Convex Sparse Model
This work provides theoretical insights for sparse signal processing and optimization, but it is incremental as it extends known results to a more general model.
The paper tackles the problem of extending the piecewise linearity property of the min-norm solution map from the convex LASSO model to a nonconvexly regularized convex sparse model called sGMC, proving that the solution map is continuous and piecewise linear with constant sparsity patterns, and proposes an algorithm to compute it in finite iterations under suitable conditions.
It is well known that the minimum $\ell_2$-norm solution of the convex LASSO model, say $\mathbf{x}_{\star}$, 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 $\mathbf{x}_{\star}(\mathbf{y},λ)$, where $\mathbf{y}$ 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 $\ell_1$-norm as sparse regularizer, but its objective function is overall-convex. Based on the geometric properties of $\mathbf{x}_{\star}(\mathbf{y},λ)$, we propose an extension of the least angle regression (LARS) algorithm, which iteratively computes the closed-form expression of $\mathbf{x}_{\star}(\mathbf{y},λ)$ in each linear zone. Under suitable conditions, the proposed algorithm provably obtains the whole solution map $\mathbf{x}_{\star}(\mathbf{y},λ)$ within finite iterations. Notably, our proof techniques for establishing continuity and piecewise linearity of $\mathbf{x}_{\star}(\mathbf{y},λ)$ 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 $(\mathbf{y},λ)$; (b) to prove piecewise linearity and piecewise constant sparsity pattern of $\mathbf{x}_{\star}(\mathbf{y},λ)$, we do not require any assumption that previous work relies on (whereas to prove some additional properties of $\mathbf{x}_{\star}(\mathbf{y},λ)$, we use a different set of assumptions from previous work).