Approximate Solutions for the Bilinear Form Computational Problem

The border rank of the multiplication of $2\times 2$ matrices is seven

“…By the linear indepence of c 3 , c 5 , c 6 and α 3 , α 5 , α 6 , the matrix must have rank two. There are two subcases to consider depending on whether or not β 3 (b) = 0 in case 1 (resp.…”

“…These perturbed maps can give rise to fast exact algorithms for matrix multiplication; see [7]. The border rank made appearances in the literature in the 1980s and early 1990s (see, e.g., [6,5,19,8,15,2,10,11,4,3,17,16,18,1,9,21]), but to our knowledge there has not been much progress on the question since then.…”

“…A related problem is to determine the border rank of matrix multiplication (or any given bilinear map), first introduced in [6,5]. Roughly speaking, some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small "good" perturbations of the map.…”

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

“…It was introduced in [3,4]. The CPD and related decompositions have numerous applications [5,6,7,8,9,10,11,12], and various iterative CPD algorithms are available [13]. Unfortunately, for R ≥ 2 the problem may not have an optimal solution because the set S R (I, J, K) is not closed [14].…”

Multivariate power series multiplication

“…We obtain two kinds of results: on the rank of the multiplication modulo an arbitrary 0-dimensional monomial ideal, and on the total complexity in some specific cases. The core of these results is to prove that power series modulo M can be approximately multiplied with a number of multiplications that equals the degree of M , using the idea of approximate algorithm introduced in [2,1,3].…”