A knowledge-driven nonconvex square-root regularization framework for joint group sparsity and structured smoothness

Wanling Xie, Hu Yang

Structured sparse learning is a leading paradigm for high-dimensional data modeling, as it enables rigorous integration of domain-validated structural priors into the learning process. Among these priors, pre-defined grouped structure and piecewise smoothness are ubiquitous in real-world applications. However, most existing structured regularization methods are built on squared loss and convex penalties, rarely jointly model these two complementary structural priors, and suffer from two key limitations. Their optimal tuning parameters are sensitive to the unknown noise level $σ$ due to the squared loss. Meanwhile, convex penalties induce inherent estimation bias and severe over-shrinkage of large magnitudes. To address these issues, we propose a knowledge-driven Nonconvex Structured Smooth Adjustment for Square-Root Regularization (NS$^3$R), which unifies the $σ$-free square-root loss with a joint nonconvex $\ell_{2,q}$ penalty ($0 < q < 1$) to simultaneously encode group sparsity and structured smoothness priors. Theoretically, we establish rigorous non-asymptotic error bounds for the NS$^3$R estimator under a mild $(2,q)$-group restricted compatibility condition.…