  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.
