2016/01/29 by J. M. Landsberg, Landsberg, J. M., Mateusz Michałek +1 · 2 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #Matrix Theory and Algorithms #Parallel Computing and Optimization Techniques #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1601.08229
openalex publication_date 2016/01/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the determinant polynomial) come in families that include representatives with normal forms. These normal forms will be useful both to develop new efficient algorithms and to prove lower complexity bounds. We derive a border rank version of the substitution method used in proving lower bounds for tensor rank. We use this border-substitution method and a normal form to improve the lower bound on the border rank of matrix multiplication by one, to 2n2- n+1. We also point out difficulties that will be formidable obstacles to future progress on lower complexity bounds for tensors because of the "wild" structure of the Hilbert scheme of points.