Norm

A norm is a mathematical function that assigns a non-negative length or size to vectors in a vector space, satisfying specific properties such as positivity, scalar multiplication, and the triangle inequality. Commonly used norms include the L1 norm, defined as the sum of the absolute values of the vector components, and the L2 norm, which is the square root of the sum of the squares of the components. Mathematically, for a vector x in R^n, the L1 norm is given by ||x||_1 = Σ|x_i| for i = 1 to n, while the L2 norm is defined as ||x||_2 = √(Σx_i^2). Norms are fundamental in optimization problems, particularly in regularization techniques such as Lasso (L1 regularization) and Ridge (L2 regularization), where they help prevent overfitting by constraining the model complexity. Norms also relate to the concept of distance in metric spaces, providing a foundation for various machine learning algorithms, including support vector machines and k-means clustering, where distance metrics are crucial for classification and clustering tasks.