2020/04/03 by Giorgio Ottaviani, Ottaviani, Giorgio, Philipp Reichenbach +1 · 1 citation
Computer Science · Mathematics · #14N07 #14Q20 #15A69 #68Q17 #Algebraic Geometry (math.AG) #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #cs.CC #math.AG #msc:14N07 #msc:14Q20 #msc:15A69 #msc:68Q17
paper · pdf · doi:10.48550/arxiv.2004.01492
46 pages; some major adjustments and additions in Section 2 (especially on castling transforms) and Section 6; further minor revisions in other sections
arxiv created 2022/07/29 · arxiv updated 2022/08/01
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will be expressed in terms of tensor (border) rank, (border) symmetric rank and the asymptotic rank of certain tensors. We introduce the multilinear rank of a tensor as well, deal with the concept of tensor equivalence and study prehomogeneous vector spaces with the castling transform. Moreover, we treat Apolarity Theory and use it to determine the symmetric rank (Waring rank) of some symmetric tensors.