  Hypergraph-based machine learning methods are now widely recognized as
important for modeling and using higher-order and multiway relationships
between data objects. Local hypergraph clustering and semi-supervised learning
specifically involve finding a well-connected set of nodes near a given set of
labeled vertices. Although many methods for local clustering exist for graphs,
there are relatively few for localized clustering in hypergraphs. Moreover,
those that exist often lack flexibility to model a general class of hypergraph
cut functions or cannot scale to large problems. To tackle these issues, this
paper proposes a new diffusion-based hypergraph clustering algorithm that
solves a quadratic hypergraph cut based objective akin to a hypergraph analog
of Andersen-Chung-Lang personalized PageRank clustering for graphs. We prove
that, for graphs with fixed maximum hyperedge size, this method is strongly
local, meaning that its runtime only depends on the size of the output instead
of the size of the hypergraph and is highly scalable. Moreover, our method
enables us to compute with a wide variety of cardinality-based hypergraph cut
functions. We also prove that the clusters found by solving the new objective
function satisfy a Cheeger-like quality guarantee. We demonstrate that on large
real-world hypergraphs our new method finds better clusters and runs much
faster than existing approaches. Specifically, it runs in few seconds for
hypergraphs with a few million hyperedges compared with minutes for flow-based
technique. We furthermore show that our framework is general enough that can
also be used to solve other p-norm based cut objectives on hypergraphs. Our
code is available \url{github.com/MengLiuPurdue/LHQD}.
