l1norm

L1 norm is a mathematical measure defined for vectors. For a vector x in R^n, the L1 norm, denoted ||x||1, is the sum of the absolute values of its components: ||x||1 = ∑_{i=1}^n |x_i|. It is a genuine norm, fulfilling non-negativity, positive definiteness, scalability, and the triangle inequality. The L1 unit ball has a cross-polytope (diamond-like) shape, which contrasts with the rounder Euclidean unit ball of the L2 norm and promotes sparsity in optimization.

In optimization, the L1 norm is notable for being non-differentiable at zero but convex, making it amenable to subgradient and proximal methods. A common application is L1 regularization or the Lasso, where problems take the form min_x ||Ax − b||2^2 + λ||x||1. The L1 penalty tends to drive small coefficients to exactly zero, yielding sparse solutions useful in feature selection and interpretability. In signal processing and compressed sensing, L1 minimization is used to recover sparse signals from undersampled measurements.

Relations to other norms include that the L1 norm is the dual of the L-infinity norm, meaning the maximum correlation with a unit L1 vector is bounded by the L-infinity norm of the dual variable. It also satisfies the inequality ||x||2 ≤ ||x||1 ≤ √n ||x||2 for x in R^n, linking it to the L2 norm while preserving its sparsity-promoting geometry.

For matrices, L1 can refer to different concepts: the vector L1 norm applied to all entries, or the induced matrix 1-norm, defined as the maximum absolute column sum. In machine learning and statistics, the vector L1 norm is the standard reference for sparsity-inducing penalties.