2015/12/25 by Ada Boralevi, Jan Draisma, Boralevi, Ada +5 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Matrix Theory and Algorithms #Polynomial and algebraic computation #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1512.08031
openalex publication_date 2015/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
While every matrix admits a singular value decomposition, in which the terms\nare pairwise orthogonal in a strong sense, higher-order tensors typically do\nnot admit such an orthogonal decomposition. Those that do have attracted\nattention from theoretical computer science and scientific computing. We\ncomplement this existing body of literature with an algebro-geometric analysis\nof the set of orthogonally decomposable tensors.\n More specifically, we prove that they form a real-algebraic variety defined\nby polynomials of degree at most four. The exact degrees, and the corresponding\npolynomials, are different in each of three times two scenarios: ordinary,\nsymmetric, or alternating tensors; and real-orthogonal versus complex-unitary.\nA key feature of our approach is a surprising connection between orthogonally\ndecomposable tensors and semisimple algebras---associative in the ordinary and\nsymmetric settings and of compact Lie type in the alternating setting.\n