2023/05/09 by Tommi Muller, Muller, Tommi, Vidit Nanda +3 · 1 citation
Mathematics · Medicine · #14C17 #14D21 #15A18 #15A69 #16G20 #Advanced Neuroimaging Techniques and Applications #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Spectral Theory (math.SP) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2305.05622
openalex publication_date 2023/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We count singular vector tuples of a system of tensors assigned to the edges of a directed hypergraph. To do so, we study the generalisation of quivers to directed hypergraphs. Assigning vector spaces to the nodes of a hypergraph and multilinear maps to its hyperedges gives a hyperquiver representation. Hyperquiver representations generalise quiver representations (where all hyperedges are edges) and tensors (where there is only one multilinear map). The singular vectors of a hyperquiver representation are a compatible assignment of vectors to the nodes. We compute the dimension and degree of the variety of singular vectors of a sufficiently generic hyperquiver representation. Our formula specialises to known results that count the singular vectors and eigenvectors of a generic tensor. Lastly, we study a hypergraph generalisation of the inverse tensor eigenvalue problem and solve it algorithmically.