L0L1

L0L1 refers to a class of regularization or sparsity-promoting methods in statistics, machine learning, and signal processing that combine an L0 penalty with an L1 penalty. The aim is to encourage sparse solutions, meaning only a small number of coefficients are nonzero, while also stabilizing estimates and reducing overfitting.

In a typical regression or learning problem, one minimizes a loss function augmented with a mixed penalty: F(x) = L(y, Xx) + λ0 ||x||0 + λ1 ||x||1, where L is a loss such as least squares, ||x||0 counts the number of nonzero components, and ||x||1 is the sum of absolute values of the coefficients. The L0 term enforces sparsity more aggressively than L1 alone, but it makes the optimization non-convex and computationally challenging, especially in high dimensions.

Optimization approaches for L0L1 problems are generally approximate. Exact solutions are often intractable, so practitioners use methods such as iterative hard thresholding, reweighted L1 minimization, convex-concave procedures, or surrogate penalties that mimic L0 with continuous approximations. These strategies seek a balance between sparsity and coefficient magnitude while maintaining computational feasibility.

Applications of L0L1 regularization appear in sparse regression, feature selection, and certain areas of signal processing and compressed sensing where strong sparsity is desirable alongside moderate coefficient control. It is related to, but distinct from, other sparsity-promoting penalties like the elastic net, which combines L1 and L2 norms rather than an L0 term.

Notes: The term L0L1 is used variably in the literature, and specific formulations may differ across studies and applications. See also L0 norm, L1 norm, sparse regularization, and related optimization techniques.