Spectral Sparsification of Hypergraphs

For an undirected/directed hypergraph G=(V,E), its Laplacian L_Gℝ^V→ℝ^V is defined such that its “quadratic form” x^⊤ L_G(x) captures the cut information of G. In particular, 1_S^⊤ L_G(1_S) coincides with the cut size of S ⊆ V, where 1_S ∈ℝ^V is the characteristic vector of S. A weighted subgraph H of a hypergraph G on a vertex set V is said to be an ϵ-spectral sparsifier of G if (1-ϵ)x^⊤ L_H(x) ≤x^⊤ L_G(x) ≤ (1+ϵ)x^⊤ L_H(x) holds for every x∈ℝ^V. In this paper, we present a polynomial-time algorithm that, given an undirected/directed hypergraph G on n vertices, constructs an ϵ-spectral sparsifier of G with O(n^3log n/ϵ^2) hyperedges/hyperarcs. The proposed spectral sparsification can be used to improve the time and space complexities of algorithms for solving problems that involve the quadratic form, such as computing the eigenvalues of L_G, computing the effective resistance between a pair of vertices in G, semi-supervised learning based on L_G, and cut problems on G. In addition, our sparsification result implies that any submodular function f 2^V →ℝ_+ with f(∅)=f(V)=0 can be concisely represented by a directed hypergraph. Accordingly, we show that, for any distribution, we can properly and agnostically learn submodular functions f 2^V → [0,1] with f(∅)=f(V)=0, with O(n^4log (n/ϵ) /ϵ^4) samples.