An Improvement on Ranks of Explicit Tensors
Abstract
We give constructions of nk x nk x n tensors of rank at least 2nk - O(n(k-1)). As a corollary we obtain an [n]r shaped tensor with rank at least 2n(r/2) - O(n(r/2)-1) when r is odd. The tensors are constructed from a simple recursive pattern, and the lower bounds are proven using a partitioning theorem developed by Brockett and Dobkin. These two bounds are improvements over the previous best-known explicit tensors that had ranks nk and n(r/2) respectively
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