Fellowship Of The Ring: Tensor Ranks And Matrix Multiplication Complexity
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Tensors are just multi-dimensional arrays. Notions of ranks and border rank abound in the literature. Tensor decompositions also have a lot of application in data analysis, physics, and other areas of science. I will survey my recent two results about tensor ranks and their application to matrix multiplication complexity. The first result relates different notion of tensor ranks to polynomials of vanishing Hessian. The second one computes the border rank of 3 X 3 permanent, which is important in the theory of matrix multiplication complexity. I will also briefly discuss the newest technique we used to achieve our results: border apolarity. Furthermore, I will survey on how this new technology helps us to compute/bound border rank of a lot of tensors of interests that were considered to be inaccessible before.
Tensor Ranks and Matrix Multiplication Complexity