  Graph sparsification has been studied extensively over the past two decades,
culminating in spectral sparsifiers of optimal size (up to constant factors).
Spectral hypergraph sparsification is a natural analogue of this problem, for
which optimal bounds on the sparsifier size are not known, mainly because the
hypergraph Laplacian is non-linear, and thus lacks the linear-algebraic
structure and tools that have been so effective for graphs.
  Our main contribution is the first algorithm for constructing
$\epsilon$-spectral sparsifiers for hypergraphs with $O^*(n)$ hyperedges, where
$O^*$ suppresses $(\epsilon^{-1} \log n)^{O(1)}$ factors. This bound is
independent of the rank $r$ (maximum cardinality of a hyperedge), and is
essentially best possible due to a recent bit complexity lower bound of
$\Omega(nr)$ for hypergraph sparsification.
  This result is obtained by introducing two new tools. First, we give a new
proof of spectral concentration bounds for sparsifiers of graphs; it avoids
linear-algebraic methods, replacing e.g.~the usual application of the matrix
Bernstein inequality and therefore applies to the (non-linear) hypergraph
setting. To achieve the result, we design a new sequence of
hypergraph-dependent $\epsilon$-nets on the unit sphere in $\mathbb{R}^n$.
Second, we extend the weight assignment technique of Chen, Khanna and Nagda
[FOCS'20] to the spectral sparsification setting. Surprisingly, the number of
spanning trees after the weight assignment can serve as a potential function
guiding the reweighting process in the spectral setting.
