Sparse

Sparse refers to data structures or signals in which most components are zero or negligibly small. In mathematics and computer science, a vector is sparse if only a small subset of its elements are nonzero; a matrix is sparse if the majority of its entries are zero. Exploiting sparsity enables substantial savings in memory and computation.

Sparse matrices are stored in specialized formats that record only nonzero values and their positions. Common formats include CSR (compressed sparse row), CSC (compressed sparse column), and COO (coordinate list). The choice of format affects the efficiency of operations such as sparse matrix–vector multiplication (SpMV) and matrix factorization. Sparse storage is especially important for large-scale problems in engineering, physics, and data science, where nonzero patterns reflect underlying structure, such as graphs or finite element meshes.

Applications span solving large sparse linear systems and eigenvalue problems, graph analytics, and machine learning with high-dimensional, sparse feature spaces. In compressed sensing, sparsity underpins the recovery of signals from limited measurements. Sparse representations are also used in NLP, image processing, and recommendation systems.

Key algorithms include direct methods (sparse Cholesky, LU) and iterative solvers (conjugate gradient, GMRES, BiCGSTAB) that are tailored to sparse data. Preconditioning is often essential to accelerate convergence. In statistics and ML, sparsity is promoted by regularization techniques such as L1 (lasso) to yield interpretable, efficient models.

Challenges include fill-in during factorization, choosing an appropriate storage format, and irregular memory access patterns that limit hardware performance. Nevertheless, sparsity remains a central principle in scalable computation and data analysis.