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Tensor Flattening

A tensor flattening arranges the entries of a tensor into a matrix by grouping its indices into row and column indices. More generally, a linear flattening is a nonzero linear transformation L from a tensor space to a space of matrices.

Let rho be the largest matrix rank of L(S) over simple tensors S. Then a linear flattening gives the border rank lower bound

R__(T)>=[(rank(L(T)))/rho],

where [x] is the ceiling function. Lovitz (2026) constructed explicit tensors whose border rank exceeds the bounds obtainable from linear flattenings.

See also

Border Rank, Matrix Rank, Tensor, Tensor Decomposition 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. "Tensor Flattening." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TensorFlattening.html

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