Spectral Sparsification of Hypergraphs

For an undirected/directed hypergraph G = ( V, E ), its Laplacian L G: ℝ 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 ∊-spectral sparsifier of G if (1 – ∊ ) x ⊺ L H ( x ) ≤ x ⊺ L G ( x ) ≤ (1 + ∊ ) 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 ∊ -spectral sparsifier of G with O ( n 3 log n / ∊ 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 nonnegative hypernetwork type submodular function can be concisely represented by a directed hypergraph of polynomial size, even if the original representation is of exponential size. Accordingly, we show that, for any distribution, we can properly and agnostically learn nonnegative hypernetwork type submodular functions with O ( n 4 log( n / ∊ )/ ∊ 4 ) samples.

Authors

• Tasuku Soma
• Yuichi Yoshida

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