15.13.4 Tensor Symmetric Rank Border Context

Tensor Symmetric Rank Border Context is the setting in which a symmetric tensor's rank is examined not as an isolated, exact value but as the limiting behavior of ranks of nearby tensors, giving rise to the related invariant known as the border rank, the smallest r such that the given tensor can be approximated arbitrarily closely by symmetric tensors of decomposition count exactly r, even when the tensor itself cannot be decomposed exactly using only r pure power terms. This context matters because the set of symmetric tensors having decomposition count at most r for a fixed r is generally not closed under limits, meaning a sequence of rank-r tensors can converge to a tensor whose own exact rank is strictly larger than r, and the border rank captures this limiting, approximative behavior precisely.

Introducing the border context alongside the exact rank structure acknowledges that rank, defined purely through exact minimality, can behave discontinuously: a small perturbation of a tensor can cause its exact rank to jump, while the border rank remains stable under such perturbations in a way the exact rank does not, making border rank a genuinely distinct and independently useful invariant.

Definition of Border Rank

Rank as a Limit of Nearby Decompositions

The border rank of a symmetric tensor T, denoted here b(T), is defined as the smallest r such that T lies in the closure, under any standard topology on the space of symmetric tensors, of the set of tensors having decomposition count at most r:

so that border rank measures the smallest term count achievable in the limit, rather than exactly.

Border Rank Never Exceeds Ordinary Rank

Because any tensor of exact decomposition count r trivially forms a constant sequence converging to itself, every tensor of decomposition count r also has border rank at most r; consequently b(T) is always less than or equal to the ordinary symmetric rank r(T), with equality holding for many but not all symmetric tensors.

Strict Inequality and Its Meaning

Tensors With Border Rank Strictly Less Than Rank

For certain symmetric tensors, particularly at higher rank n and specific dimensions d, the border rank is strictly smaller than the exact decomposition count, meaning such a tensor can be approximated arbitrarily well by sequences of pure-power decompositions using fewer terms than are needed for any exact decomposition, even though no exact decomposition with that smaller number of terms exists.

The Mechanism Behind the Gap

This gap typically arises because a sequence of decompositions with a fixed, smaller term count can approach the target tensor while the individual vectors or scalars within that sequence diverge to infinity in a compensating way, a phenomenon where terms individually grow unboundedly large while their sum remains bounded and converges to the target; such degenerating sequences illustrate why the closure defining border rank can include tensors unreachable by any single finite exact decomposition of the same term count.

Border Rank and Semicontinuity

Upper Semicontinuity of Rank

The gap between border rank and exact rank reflects a general phenomenon: the exact decomposition count, viewed as a function on the space of symmetric tensors, is upper semicontinuous rather than continuous, meaning it can only jump upward, never downward, in the limit; a sequence of tensors with bounded decomposition count can converge to a tensor whose exact rank is larger, but never to one whose exact rank is smaller than the limiting behavior would suggest.

Border Rank as the Continuous Lower Envelope

Border rank can be understood as the largest lower semicontinuous function bounded above by the exact rank, capturing exactly the stable, non-jumping part of the rank invariant and discarding the discontinuous jumps that make exact rank sensitive to arbitrarily small perturbations of the tensor.

Practical Significance of the Border Context

Stability Under Approximation

Because border rank does not jump under small perturbations in the way exact rank can, it serves as a more robust invariant in settings where a symmetric tensor is known only approximately, such as when derived from noisy or finite-precision numerical data, since the border rank of a nearby, exactly measured tensor provides a stable estimate unaffected by the instability exact rank can exhibit near special, lower-rank tensors.

Border Rank Within the Decomposition Structure

Situated within the broader symmetric decomposition structure, the border context supplies the necessary refinement acknowledging that the decomposition relation, viewed as an equation to be solved exactly, has a subtly different solvability landscape than the same equation viewed as one to be solved only in the limit, with border rank precisely capturing the boundary between these two notions of solvability.