L1normi

L1normi, commonly called the L1 norm, is a norm on R^n defined for a vector x = (x1, ..., xn) by ||x||1 = sum_i |xi|. It satisfies positivity, homogeneity, and the triangle inequality, and it provides a measure of the magnitude of a vector based on the sum of the magnitudes of its components.

The L1 norm induces a distance (a metric) between two vectors x and y given by d1(x, y) = ||x − y||1 = sum_i |xi − yi|. Geometrically, the unit ball {x : ||x||1 ≤ 1} is a cross-polytope; in 2D it appears as a diamond, and in higher dimensions as a diamond-like polytope with many facets. The L1 norm is part of the family of p-norms, with p = 1, and its dual norm is the L∞ norm, defined by ||y||∞ = max_i |yi|.

In optimization and statistics, the L1 norm is valued for promoting sparsity when used as a regularization term (for example, in the LASSO). It is also used in robust regression and compressed sensing. Computationally, the L1 norm is non-differentiable at points where xi = 0 for some i, which influences optimization algorithms. Projections onto the L1 ball can be computed efficiently via sorting and thresholding (a method described in algorithms for projecting onto L1 constraints).

Examples: for x = (3, −4, 0), ||x||1 = 7. The distance between a and b, say (1, 2) and (4, −1), is |3| + |−3| = 6.

See also: Lp norms, L2 norm, L∞ norm, sparse recovery.