Tensor unfoldings and mode ranks

For a three-way array of size n₁ by n₂ by n₃, the first unfolding has rows i and columns indexed by (j, k). Use j as the fast index, so column number is j + n₂(k − 1), with indices starting at one. In the second unfolding the remaining indices are (i, k), with i fast; in the third they are (i, j), again with i fast. For more modes, list remaining modes in increasing order and let the first listed index vary fastest.

Other books and software use different valid orders. Changing only the column order multiplies the unfolding by a permutation matrix, preserving singular values and matrix rank. However, formulas involving Kronecker or Khatri–Rao products must use the matching convention. A dimensionally correct product can still reconstruct permuted entries if its ordering is wrong. Writing one small slice example before deriving a large formula is an effective way to catch that error.

The multilinear rank is the tuple rₙ = rank(T₍ₙ₎). Each component counts the dimension of the span of fibers in its mode. These ranks can differ: a tensor that uses one third-mode direction may still contain a full-rank matrix in the first two modes. Every nonzero pure tensor has all mode ranks one, and conversely a nonzero tensor with every mode rank one is pure.

If a tensor is a sum of R pure products, each unfolding is a sum of R rank-one matrices. Therefore every mode rank is at most R, giving a lower bound on CP rank. The tuple does not in general determine CP rank; later we will see a two-by-two-by-two tensor with mode ranks (2, 2, 2) but CP rank three. Matrix singular values supply useful bounds without collapsing all notions of tensor rank into one.

Exact rank changes discontinuously when a singular value becomes exactly zero. Numerical rank instead depends on a tolerance and a selected norm or error budget. A small singular value may be negligible relative to measurement noise while remaining algebraically nonzero. Report the threshold or discarded Frobenius tail rather than declaring a tensor low rank without qualification. Scaling the entire data array changes absolute tolerances but not exact ranks.

Modewise low-rank approximations select subspaces for different indices. Their projections can be combined because they act on distinct modes, leading to Tucker compression. A single chosen unfolding may compress one direction well and leave the remaining structure large. Inspecting the full rank tuple and tail spectra reveals where storage reduction is possible and where a proposed mode size is still an essential information bottleneck.