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Border Rank

The border rank R__(T) of a tensor T over the complex numbers is the least integer r such that T is a limit of tensors of tensor decomposition rank at most r. Equivalently, T lies in the closure of the set of tensors expressible as a sum of at most r simple tensors. Border rank can be smaller than tensor decomposition rank (Lovitz 2026); neither is the index-counting meaning of tensor rank.

Lovitz (2026) reported an explicit family of tensors T_n in C^n tensor C^n tensor C^n with integer entries and

R__(T_n)>=3n-11n^(3/4).

The entries are computable in polynomial time in n. Lovitz (2026) also showed that linear tensor flattening methods cannot establish a border-rank lower bound exceeding 2n-1, with that ceiling attained when n is odd.

Lovitz (2026) states that GPT-6 Astra obtained proofs that were subsequently verified and simplified.

See also

Tensor, Tensor Decomposition Rank, Tensor Flattening, Tensor Rank

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References

Lovitz, B. "Explicit Tensors Beyond the Linear Flattening Barrier." 6 Oct 2026. https://arxiv.org/abs/2610.08504.

Cite this as:

Weisstein, Eric W. "Border Rank." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BorderRank.html

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