L1normen

L1Normen refers to the class of norms based on the L1 measure, also known as the "taxicab" or "Manhattan" norm. These norms are used in various mathematical and applied fields, including linear algebra, statistics, and machine learning, to quantify the size or magnitude of vectors or matrices.

The L1 norm of a vector x = (x₁, x₂, ..., xₙ) in n-dimensional space is defined as the sum of the absolute values of its components: ||x||₁ = |x₁| + |x₂| + ... + |xₙ|. It measures the total "taxicab" distance traveled along axes from the origin to the point x, aligning with the geometry of the city grid.

L1 norms are notable for promoting sparsity in solutions when used as regularization terms in optimization problems, such as in Lasso regression. This property encourages models to select a subset of relevant features by shrinking some coefficients precisely to zero.

The L1 norm exhibits several mathematical properties: it is a convex function, positively homogeneous, and satisfies the triangle inequality. These properties class it among the class of vector norms, making it suitable for various analytical and computational methods.

Extensions of L1 norms also exist for matrices, known as the nuclear norm or the matrix L1 norm, which measure different aspects of matrix magnitude. Additionally, L1-based metrics are employed in compressed sensing, robust statistics, and image processing for their ability to handle outliers and enforce sparsity.

Overall, L1Normen form a fundamental concept in the mathematical toolkit for analyzing and solving problems involving vector and matrix magnitudes, especially when sparsity and robustness are desired features.