w21Lnorm

w21Lnorm stands for the weighted l21 norm, a convex norm used as a regularizer in optimization to promote group sparsity. It operates on a vector x ∈ R^n that is partitioned into disjoint groups G_1, G_2, ..., G_m. With nonnegative group weights w_j ≥ 0, the w21 norm is defined as ||x||_{w21} = sum_{j=1}^m w_j ||x_{G_j}||_2, where x_{G_j} denotes the subvector of x restricted to group G_j.

As a regularizer, the w21 norm encourages whole groups of coefficients to be driven to zero, enabling structured sparsity. When all weights are positive, ||·||_{w21} is a norm; if any weight is zero it becomes a seminorm. The norm is monotone with respect to group shrinkage and is homogeneous and subadditive, preserving many analytical properties useful in optimization.

The w21 norm is a generalization of the group lasso. If all groups are treated equally (w_j equal) and the groups partition the coordinates, it reduces to the standard group lasso penalty. In practice, it is employed in problems where features are naturally grouped, such as multi-task learning, genomics, image analysis, and signal processing, to select relevant groups rather than individual features.

Computationally, solving regularized problems with ||x||_{w21} often uses proximal methods. The proximal operator acts group-wise, performing a shrinkage (group soft-thresholding) on each x_{G_j} by a factor dependent on w_j and the step size. This yields efficient, scalable algorithms for large-scale problems.