Cactus barriers

Determinantal methods for bounding the rank and border rank of tensors or polynomials are subject to a major barrier. For instance, it is known that using determinantal methods one cannot prove a lower bound for the border rank of a 3-way tensor of size m in each direction that exceeds 6m-4. We explain the precise geometric reason for this number (and analogous bounds in more general tensor spaces) using cactus varieties and, more generally, scheme theoretic methods in algebraic geometry.

💡 Research Summary

The paper “Cactus Barriers” investigates why determinantal (matrix‑based) techniques for proving lower bounds on tensor rank and border rank inevitably hit a universal ceiling. The classical approach, often called a “linear rank method,” selects a matrix M of linear forms on a vector space W and studies the locus {rk M ≤ k}. If a variety X of simple tensors (or polynomials) is contained in this rank‑k locus, then for any tensor F one obtains the inequality r_X(F) ≥ ⌈rk M(F)/k⌉, giving a lower bound for the X‑rank and, by the same argument, for the X‑border rank.

The authors show that this method is fundamentally limited by the geometry of cactus varieties. For a projective variety X⊂P(W) with smooth locus X₀, they define the usual secant varieties σ_r(X) (spanned by r points of X) and the cactus secant varieties K_r(X) (spanned by finite schemes of length r supported on X). The key result, Theorem 1.3, states that if X⊂{rk M ≤ k} then K_r(X₀)⊂{rk M ≤ k·r}. Consequently, any determinantal method can only detect the cactus rank, never the ordinary border rank, unless the two coincide.

If there exists an integer g such that K_g(X₀)=P(W) (i.e., the generic cactus rank equals g), then no matrix M can ever give a lower bound on the X‑border rank larger than g. This explains the well‑known “6m‑4 barrier” for 3‑way tensors in C^m⊗C^m⊗C^m: the generic cactus rank of the Segre variety P^{m‑1}×P^{m‑1}×P^{m‑1} is exactly 6m‑4, so any linear rank method is capped at this value.

The paper surveys many concrete families (Veronese, Segre, Segre‑Veronese, etc.) and observes that while the generic X‑rank is typically on the order of dim W·dim X+1, the generic cactus rank is dramatically smaller. For example, for the Segre embedding of P^{a‑1}×P^{b‑1}×P^{c‑1} the cactus rank needed to fill the ambient space is g=2(a+b+c‑2), whereas the ordinary secant variety requires roughly abc‑a‑b‑c+2 points. This disparity grows rapidly with the dimensions, producing huge gaps between the best known lower bounds (derived from determinantal methods) and the true border rank.

The authors place their result in context with earlier partial barriers (e.g.,