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

• Published in: IEEE transactions on information theory Vol. 71; no. 11; p. 1
• Main Authors: Zhang, Yi, Yamada, Isao
• Format: Journal Article
• Language: English
• Published: IEEE 01.11.2025
• Subjects: Analytical models; compressed sensing; Cost function; generalized minimax concave penalty; Geometry; Hands; least angle regression; Linearity; nonconvexly regularized convex models; Reviews; Shape; Signal processing algorithms; Sparse least-squares problems; Sparse matrices; Training
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2025.3605554

It is well known that the minimum ℓ 2 -norm solution of the convex LASSO model, say x ⋆ , is a continuous piecewise linear function of the regularization parameter λ, and its signed sparsity pattern is constant within each linear piece (Osborne 2000, Efron et al. 2004). The current study is an extension of this classic result, proving that the aforementioned properties extend to the min-norm solution map x ⋆ ( 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 ℓ 1 -norm as sparse regularizer, but its objective function is overall-convex. Based on the geometric properties of x ⋆ ( y , λ), we propose an extension of the least angle regression (LARS) algorithm, which iteratively computes the closed-form expression of x ⋆ ( y , λ) in each linear zone. Under suitable conditions, the proposed algorithm provably obtains the whole solution map x ⋆ ( y , λ) within finite iterations. Numerical experiments demonstrate the efficiency and reduced estimation error of the proposed algorithm compared to the conventional LARS. Notably, our proof techniques for establishing continuity and piecewise linearity of x ⋆ ( 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 x ⋆ ( y , λ), we do not require any assumption that previous work relies on (whereas to prove some additional properties of x ⋆ ( y , λ), we use a different set of assumptions from previous work).