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9 votes

What is expected (border) rank of the knonecker product of 3-tensors

We don't know. There are formats in which the equality is false for generic tensors: take $F^{n \times n \times 1}$ and $F^{n \times n \times n^2}$. Generic tensors in both formats are isomorphic to m …
Vladimir Lysikov's user avatar
3 votes

What is the best known bound for the bilinear complexity of $4\times 4$ matrices product

In the paper you mention, Bläser improves the lower bound to 34. To my knowledge, there was no progress on this specific problem. The recent Nature paper which used machine learning to solve the usual …
Vladimir Lysikov's user avatar
5 votes

How far is the slice rank of a tensor from its CP rank

No. For $n \times n \times n$ tensors slice rank does not exceed $n$, while tensor rank can be as large as $\frac{n^3}{3n - 2} \sim \frac{n^2}{3}$ by dimension count.
Vladimir Lysikov's user avatar