sparsification

Sparsification is the process of reducing the number of nonzero elements in a data structure, such as a graph’s adjacency matrix, or in a matrix arising from a computation, while preserving essential properties of the original object. The goal is to obtain a sparser representation that supports efficient computation without significantly compromising accuracy. Sparsification appears in several domains, including graph theory, numerical linear algebra, signal processing, and machine learning. In practice, sparsifiers aim to maintain key measures like cuts, spectra, or other quadratic forms.

In graph theory, a sparsifier is a sparse graph that approximates the original graph with respect to certain properties. A cut sparsifier preserves the values of all edge cuts up to a factor (1±ε). A spectral sparsifier preserves the Laplacian quadratic form, so x^T L_G x is approximated by x^T L_H x for all vectors x. Classic results show that dense graphs can have sparse sparsifiers with O(n log n / ε^2) edges, enabling faster algorithms. Methods include random edge sampling proportional to edge strength (Benczúr-Karger) and sampling guided by effective resistance (Spielman-Srivastava). Reweighting of kept edges is often used to improve accuracy.

In numerical linear algebra, sparsification replaces dense matrices with sparse approximations to accelerate linear solves and eigenvalue computations, often as part of preconditioning. In machine learning and signal processing, sparsification appears in model compression and sparse representations, such as pruning neural networks or selecting a subset of features or coefficients while preserving predictive performance or signal fidelity. Sparsification is typically probabilistic, carrying guarantees that hold with high probability as a function of ε and graph size.

Limitations include potential loss of accuracy, dependence on the structure of the input, and computational cost of constructing the sparsifier. The choice of what to preserve—cuts, spectrum, or other properties—determines the method and guarantees provided.