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Computing symmetric rank for symmetric tensors
A. Bernardi, A. Gimigliano, M. Idà
TL;DR
The paper asks how symmetric rank can be determined and organized geometrically for symmetric tensors. It uses algebraic geometry to develop rank algorithms and describe rank strata in selected Veronese secant varieties. The results include a faster dimension-two algorithm and analyses of tensors with small border rank, while general stratum descriptions and efficient equation methods remain limited.
Problem
Determining symmetric rank and its geometric strata is difficult because defining equations for secant varieties are often insufficient, although such equations would enable border-rank computation.
Method
The paper applies projective and algebraic geometry to Veronese secant varieties, combining rank algorithms with analyses of schemes and low-border-rank cases.
Results
For dim(V)=2, a new Sylvester-algorithm version computes symmetric rank without an explicit decomposition and improves speed; the paper also describes selected rank strata.
Takeaways & Limitations
Symmetric rank can be computed or stratified for several low-dimensional and small-border-rank cases using geometric structure.
Takeaways & Limitations
A direct method for obtaining secant-variety equations is computationally inefficient and works only in very simple cases, while general rank-stratum descriptions remain unavailable.
Abstract
from arXiv · showhide
We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for $2\times ... \times 2$ tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.
1. Introduction
The paper frames symmetric tensor rank through algebraic geometry, connecting it to applications and the Waring problem. It develops faster rank computation in dimension two and studies rank strata within selected Veronese secant varieties.
- Motivation: Symmetric tensor rank matters in antenna processing, telecommunications, statistics, and data analysis, and extends SVD for symmetric matrices.These applications motivate representing symmetric tensors and determining their rank.
- Waring formulation: The Big Waring Problem asks for the minimum number of d-th powers of linear forms representing a generic degree-d form, equivalently its symmetric rank.Symmetric tensors correspond to homogeneous polynomials under the Veronese construction.
- Geometric gap: Although secant-variety dimensions are known, defining equations are often incomplete, limiting computation of symmetric border rank.Catalecticant equations are generally insufficient to describe all secant-variety ideals.
- Contributions: For dim(V)=2, the paper presents a faster Sylvester-algorithm variant that computes symmetric rank without constructing an explicit decomposition.Avoiding decomposition improves algorithmic speed, while finding explicit decompositions remains open.
- Contributions: The paper describes symmetric-rank strata for σr(X1,d), σ2(Xn,d), σ3(Xn,d), and σr(X2,4) for r≤5, with rank algorithms in the first three cases.It also studies rank on σ2 of elliptic normal curves.
2. Preliminaries
The preliminaries define symmetric rank, border rank, Veronese varieties, secant varieties, and rank strata. They establish scheme-theoretic tools for analyzing secant membership while noting limits on general geometric descriptions.
- Rank notions: A symmetric tensor has rank r when it is the sum of r, and no fewer, symmetric rank-one tensors.Rank-one tensors have the form v⊗d.
- Veronese varieties: The Veronese variety Xn,d parameterizes projective classes of symmetric rank-one tensors and d-th powers of linear forms.It is the degree-d Veronese embedding of projective space.
- Secant varieties: The s-th secant variety is generated by spans of s points of X, and membership defines symmetric border rank.The symmetric rank locus is not closed, motivating secant varieties and border rank.
- Scheme-theoretic criterion: Under the stated Hilbert-scheme assumptions, P belongs to σr(X) exactly when a degree-r zero-dimensional scheme Z⊂X spans a projective space containing P.The proposition is applied repeatedly to Veronese varieties, including cases where the Hilbert scheme is irreducible and imposes independent conditions.
- Counterexample: For X2,6, an 8-point scheme on a line spans only a P6, showing that a linear space in σr(X) need not be spanned by a degree-r scheme of X.The example distinguishes containment in a secant variety from representation by a degree-r spanning scheme.
- Rank strata: The loci σb,r(Xn,d) collect tensors with border rank b and symmetric rank r, but their general geometric description is not known.The authors expect these loci to be locally closed over an algebraically closed field but lack a general reference.
3. Two dimensional case
For two-dimensional V, the Veronese variety is a rational normal curve, enabling geometric rank characterizations and algorithms based on secant spaces and catalecticant matrices. The section also extends the rank-stratification perspective to elliptic normal curves.
- Sylvester algorithm: The simplified Sylvester algorithm computes symmetric rank without explicitly producing the tensor decomposition, improving speed by avoiding decomposition construction.The full procedure instead outputs a sum of powers of linear forms, while the simplified version returns only the minimal rank.
- Geometric description: For dim(V)=2, symmetric rank is the minimum secant-space dimension whose span contains the tensor and meets the rational normal curve in distinct points.The rational normal curve C_d parameterizes decomposable symmetric tensors.
- Sylvester algorithm: Catalecticant matrices provide the rank test underlying the two-variable Sylvester procedure, with kernel polynomials whose distinct roots identify decomposition points.When r≤d/2, a nontrivial kernel condition is equivalent to vanishing maximal minors of the catalecticant matrix.
- Geometric description: The maximum symmetric rank on the rational normal curve is d, while forms in the relevant high-rank strata have rank d−r+2; monomials x^(d−s)y^s are included.A generic hyperplane through a tensor cuts d distinct points on the curve, establishing attainability of rank d.
- Geometric description: For rational normal curves, high-rank strata inside secant varieties are described by tangential joins: σ2,d(C_d)=τ(C_d)\C_d, and σr,d−r+2(C_d)=J(τ(C_d),σr−2(C_d))\σr−1(C_d).The stated formula applies for d>2 and 3≤r<(d+2)/2.
- Elliptic normal curves: For elliptic normal curves, secant points outside the curve have rank 2 or d−1 when d≥4, whereas in Γ4⊂P3 rank-3 points are characterized by intersections of two tangent lines.The quartic case decomposes σ2(Γ4)\Γ4 into rank-2 and rank-3 strata.
4. Beyond dimension two
The paper extends its algebraic-geometric analysis from binary forms to secant varieties of Veronese varieties, giving rank algorithms and explicit symmetric-rank strata in several low-border-rank cases.
- Rank-two secants: Any tensor in σ2(Xn,d) has symmetric rank only 1, 2, or d, and an algorithm tests membership and distinguishes ranks 2 and d.This gives a complete rank classification for the second secant variety in arbitrary dimension.
- Secant schemes: A zero-dimensional scheme of degree at most 2d + 1 imposes independent degree-d conditions exactly when no line meets it with degree at least d + 2.This criterion is used in the geometric analysis of small secant schemes.
- Third secants: For σ3(Xn,d) \ σ2(Xn,d), the rank strata are explicitly classified: three strata for d = 3 and four strata for d ≥ 4.For d = 3 the strata are σ3,3, σ3,4, and σ3,5; for d ≥ 4 they are σ3,3, σ3,d−1, σ3,d+1, and σ3,2d−1.
- Secant varieties of X2,3: For cubic forms in three variables, the complete stratification includes ranks 1, 2, 3, 4, and 5 across X2,3, its secant differences, and the complement of σ3(X2,3).The complement P9 \ σ3(X2,3) is the rank-4 stratum σ4,4(X2,3).