Nonzeros

Nonzeros are elements in a mathematical object that are not equal to zero. The term is used across algebra, analysis, and combinatorics to distinguish meaningful values from zeros. For a vector x in R^n, the nonzero entries are the coordinates with x_i ≠ 0. The set of indices where x_i ≠ 0 is called the support of x, and the number of nonzero entries is denoted ||x||_0, a function that counts nonzeros but is not a norm in the mathematical sense.

In a matrix, nonzero entries similarly indicate activity; the term sparsity describes having many zeros. Sparse matrices are stored and processed efficiently by data structures that only record nonzero entries and their locations, using formats such as compressed sparse row or compressed sparse column.

Counting nonzeros is a standard operation in algorithms for graphs, networks, and data analysis, where the support of a vector or column can indicate which components are active. The notion of nonzeros is also central to the study of sparsity, which describes how many nonzero elements a dataset or model contains.

In statistics and machine learning, sparsity refers to models with few nonzero coefficients, enabling interpretability and efficiency. The L0 “norm” counts nonzeros but is not a true norm; it is often relaxed to the L1 norm to encourage sparsity in optimization problems.

In sequences and functions, a nonzero value contributes to sums, transforms, and evaluations; for example, the sequence (0, 4, 0, -2) has two nonzeros. Related concepts include zero elements, support, and sparsity.