Spectral Sparsification of Hypergraphs

For an undirected/directed hypergraph $G=(V,E)$, its Laplacian $L_G\colon\mathbb{R}^V\to \mathbb{R}^V$ is defined such that its ``quadratic form'' $\boldsymbol{x}^\top L_G(\boldsymbol{x})$ captures the cut information of $G$. In particular, $\boldsymbol{1}_S^\top L_G(\boldsymbol{1}_S)$ coincides with the cut size of $S \subseteq V$, where $\boldsymbol{1}_S \in \mathbb{R}^V$ is the characteristic vector of $S$. A weighted subgraph $H$ of a hypergraph $G$ on a vertex set $V$ is said to be an $\epsilon$-spectral sparsifier of $G$ if $(1-\epsilon)\boldsymbol{x}^\top L_H(\boldsymbol{x}) \leq \boldsymbol{x}^\top L_G(\boldsymbol{x}) \leq (1+\epsilon)\boldsymbol{x}^\top L_H(\boldsymbol{x})$ holds for every $\boldsymbol{x} \in \mathbb{R}^V$. In this paper, we present a polynomial-time algorithm that, given an undirected/directed hypergraph $G$ on $n$ vertices, constructs an $\epsilon$-spectral sparsifier of $G$ with $O(n^3\log n/\epsilon^2)$ 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 $f\colon 2^V \to \mathbb{R}_+$ with $f(\emptyset)=f(V)=0$ can be concisely represented by a directed hypergraph. Accordingly, we show that, for any distribution, we can properly and agnostically learn submodular functions $f\colon 2^V \to [0,1]$ with $f(\emptyset)=f(V)=0$, with $O(n^4\log (n/\epsilon) /\epsilon^4)$ samples.