On the Geometry of Border Rank Decompositions for Matrix Multiplication and Other Tensors with Symmetry

Abstract

Abstract. We present a new approach to study tensors with symmetry, via local algebraic geometry. Border rank decompositions for such tensors-in particular, matrix multiplication and the determinant polynomial-come in families. We prove that these families include representatives with normal forms. These normal forms will be useful to prove lower complexity bounds and possibly even to determine new decompositions. We derive a border rank version of the substitution method used in proving lower bounds for tensor rank. Applying these methods, we improve the lower bound on the border rank of matrix multiplication. 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.

Probabilistic Tensors and Opportunistic Boolean Matrix Multiplication

“…Landsberg [53]). Recent work in this direction in the context of small tensors includes Chiantini, Ikenmeyer, Landsberg, and Ottaviani [24], Ballard, Ikenmeyer, Landsberg, and Ryder [10], Landsberg and Micha lek [54], and Landsberg and Ryder [55].…”

Computing images of polynomial maps

“…Tensors represent multi-linear maps. Good representations of a tensor, for instance its rank decomposition, yield algorithms of lower complexity [18,19].…”

“…The multiplication of two n × n matrices is a bilinear operation and thus is represented by a -dimensional tensor. The complexity of the optimal algorithm for multiplying matrices is known to be governed by the rank (or border rank) of the associated tensor, see [18,19]. (Let us point out that it is not known in general if the rank and the border rank of the matrix multiplication tensor coincide.…”

Plethysm and fast matrix multiplication

“…In terms of algebra we know that this question is equivalent to estimating rank or border rank of a specific tensor M n,n,n ∈ C n 2 ⊗ C n 2 ⊗ C n 2 [1,8,9]. The current best lower and upper bounds are presented in [10,[12][13][14]20].…”