CP decomposition, border rank, and missing minimizers

A CP decomposition writes a tensor as a sum of R pure products. Its CP rank is the smallest such R over the chosen field. Border rank is the smallest R for which the tensor lies in the ordinary Euclidean closure of tensors with CP rank at most R. Thus border rank is at most CP rank. Over complex arrays use the real topology of their real and imaginary coordinates; real and complex ranks can differ, so the field must always be recorded.

Changing between finite-dimensional norms does not change which sequences converge, but our quantitative errors use Frobenius norm. A fixed rank-at-most-R CP set need not be closed for order three or higher. This is different from saying that one parameterization is redundant: nonclosedness concerns represented tensors themselves, after all equivalent factor scalings have been identified.

Let e₁ and e₂ be the standard basis of real two-space. The tensor W with one e₂ in each of three possible positions is visibly a sum of three pure tensors. Its mode ranks are all two, but those lower bounds alone do not prove rank three. For that stronger conclusion examine the third-index slices S = [[0, 1], [1, 0]] and D = [[1, 0], [0, 0]].

If a two-term CP representation existed, the first two factor matrices would be invertible because the corresponding mode ranks are two. Since S is invertible, its two slice coefficients would both be nonzero. Consequently DS⁻¹ would be similar to a diagonal matrix of coefficient ratios. Yet DS⁻¹ = [[0, 1], [0, 0]] is nonzero nilpotent and cannot be diagonalized. This contradiction proves CP rank three over R, and the same argument works over C.

Expansion gives Wε − W equal to ε times the three coordinate terms having two e₂ factors, plus ε² times e₂ ⊗ e₂ ⊗ e₂. These four coordinate tensors are orthonormal, so the squared Frobenius error is 3ε² + ε⁴. Each displayed CP component has magnitude proportional to 1/|ε|, illustrating cancellation rather than a bounded convergent factorization. Small residual alone cannot diagnose stable components. An unfolding of W has rank two, while the matrix rank-at-most-one set is closed. Therefore W cannot be a limit of pure tensors, closing the lower bound: its border rank is exactly two.

Because W has rank three but is a limit of rank-at-most-two tensors, its best rank-at-most-two approximation has infimum error zero and no minimizer. This exact example does not assert that every CP fitting problem lacks a solution. Rank-one approximation is better behaved, and constraints or penalties can restore compactness in particular models. Such changes alter the feasible problem; they should not be mistaken for a general higher-order Eckart–Young theorem.