l0norm

l0norm, written as ||x||_0, is defined for a vector x in R^n as the number of its nonzero components. It is commonly called the zero-norm, but it is not a true norm: it is not homogeneous (||αx||_0 = ||x||_0 for any nonzero α) and it is not convex; it is also discontinuous at points where components cross zero. Some authors extend the idea to matrices by counting nonzero entries.

In signal processing and machine learning, ||x||_0 serves as a sparsity measure and as a hard penalty in optimization. A typical objective is min ||Ax - b||_2^2 subject to ||x||_0 ≤ k, or min ||Ax - b||_2^2 + λ||x||_0. Since l0 minimization is combinatorial and NP-hard, practical methods rely on relaxations or approximations: convex relaxation replaces ||x||_0 with ||x||_1 (basis pursuit, Lasso), nonconvex penalties, iterative hard thresholding, and greedy algorithms like matching pursuit or orthogonal matching pursuit.

The l0-norm is central to sparse representation and compressed sensing. Under suitable conditions on the sensing matrix A (for example, restricted isometry properties) and sufficient sparsity, accurate recovery is possible with certain algorithms, even in the presence of noise. It underpins feature selection, dictionary learning, and sparse coding by providing a formal measure of sparsity and guiding sparsity-promoting techniques.