Length1

Length1 is an informal term used in mathematics and computer science to refer to the L1 length or L1 norm of a vector. For a vector x in n dimensions, the Length1 norm is defined as the sum of the absolute values of its components: ||x||1 = sum over i of |x_i|. The corresponding distance between two vectors x and y is the L1 distance, d1(x, y) = ||x − y||1.

Mathematically, Length1 is one of the family of Lp norms, where p ≥ 1. It satisfies the norm properties: non-negativity, positive definiteness, scalability, and the triangle inequality. It is continuous and convex, but it is not differentiable at any coordinate where x_i = 0. The unit ball under the Length1 norm in R^n is a cross-polytope (an octahedron in three dimensions).

Geometric interpretation and comparisons are common in discussion of Length1. The L1 unit ball’s shape leads to sparsity-promoting behavior in optimization problems, contrasting with the smooth, round unit ball of the L2 norm. This makes Length1 favored in contexts that reward sparse solutions.

Applications of Length1 are widespread. In statistics and machine learning, L1 regularization (also called Lasso) adds a penalty proportional to ||β||1 to encourage sparse models. In signal processing and compressed sensing, L1 minimization is used to recover sparse signals from limited or noisy measurements. The L1 loss, which minimizes the sum of absolute residuals, is more robust to outliers than the L2 loss.

In practice, Length1 can be computed efficiently and is commonly implemented in optimization algorithms, often reformulated as a linear program or handled via subgradient methods due to nondifferentiability at zero.