  Capturing complex high-order interactions among data is an important task in
many scenarios. A common way to model high-order interactions is to use
hypergraphs whose topology can be mathematically represented by tensors.
Existing methods use a fixed-order tensor to describe the topology of the whole
hypergraph, which ignores the divergence of different-order interactions. In
this work, we take this divergence into consideration, and propose a
multi-order hypergraph Laplacian and the corresponding total variation. Taking
this total variation as a regularization term, we can utilize the topology
information contained by it to smooth the hypergraph signal. This can help
distinguish different-order interactions and represent high-order interactions
accurately.
