15.14.4 Tensor Symmetric Decomposition Rank Relation

Tensor Symmetric Decomposition Rank Relation is the body of results that describes how the ordinary (non-symmetric) tensor rank of a symmetric tensor compares to its symmetric rank, and how these two notions of rank interact under the operations that produce a symmetric decomposition of a tensor.

A symmetric tensor of order d over a vector space V admits, generically, a decomposition as a sum of d-th powers of vectors,

where each term is the d-th tensor power of a vector v_i belonging to V. The smallest number of terms, r, for which such a decomposition exists is called the symmetric rank of T, written as rank_S(T). Independently, T can also be regarded simply as an element of the larger space of all tensors of the same order, and there the ordinary tensor rank, rank(T), is defined as the smallest number of simple (rank-one, not necessarily symmetric) tensors needed to write T as a sum. The Rank Relation is precisely the comparison

together with the precise conditions under which equality holds, and the description of cases in which the inequality is strict.

Foundational Framework

Symmetric Tensors as a Subvariety

The space of symmetric tensors of order d on an n-dimensional vector space, denoted by the symbol S^d V, sits inside the full tensor power of V taken d times as the subspace fixed by the action of the symmetric group permuting the tensor factors. Any decomposition of T as a sum of rank-one tensors in the full tensor power space is a decomposition in the ambient, unrestricted sense; a symmetric decomposition additionally requires every rank-one summand to itself be symmetric, i.e., of the form of a d-th tensor power of a single vector.

Two Notions of Minimality

Because every symmetric decomposition is automatically a valid (non-symmetric) decomposition, the set of achievable ranks under the symmetric constraint is a subset of the set of achievable ranks under the unrestricted constraint. Minimizing over a smaller admissible set can only produce an equal or larger minimum, which yields immediately the inequality

for every symmetric tensor T. The central question addressed by the Rank Relation is how large the gap between the two sides can become, and under what structural conditions the gap vanishes.

Comon's Conjecture and Its Status

Statement

For a long time it was conjectured, in what is known as Comon's Conjecture, that the two ranks always coincide:

for every symmetric tensor T, regardless of order or dimension.

Counterexamples

Comon's Conjecture was disproved by exhibiting symmetric tensors for which the ordinary rank is strictly smaller than the symmetric rank. The known counterexamples occur for tensors of order at least three over the complex numbers, in cases where the ambient dimension and order combine to allow decompositions built from a "hidden" non-symmetric structure that cannot be symmetrized without increasing the number of terms. These examples show that the inequality

is possible, though the gap in known examples is typically small relative to the ranks involved.

Regimes Where Equality Is Guaranteed

Despite the existence of counterexamples in general, equality between the two ranks is provable in several important regimes:

• When the order d equals two, i.e., for symmetric matrices, where rank and symmetric rank always coincide over an algebraically closed field of characteristic zero.
• When the rank is small relative to the number of variables, since low-rank decompositions of generic or subgeneric symmetric tensors can be symmetrized without cost.
• When the tensor is "generic" within its symmetric rank stratum, since the locus where the two ranks differ is contained in a proper, typically thin, subvariety of the space of symmetric tensors.

Structural Mechanisms Behind the Relation

Apolarity and the Catalecticant Method

The symmetric rank of T can be studied through apolarity theory, which associates to T an ideal of forms vanishing on it (the apolar ideal) and relates minimal symmetric decompositions to sets of points, called apolar sets, whose defining ideal is contained in the apolar ideal. The catalecticant matrices built from T furnish lower bounds for the symmetric rank via their ranks, and these bounds interact with the classical, non-symmetric flattenings used to lower-bound the ordinary tensor rank. The Rank Relation can therefore be partly understood as a comparison between the ranks of symmetric flattenings and the ranks of the full, non-symmetrized flattenings of the same tensor.

Border Rank Considerations

The relation extends naturally to border rank, the semicontinuous relaxation of rank obtained by allowing decompositions to be approximated by limits. The symmetric border rank and the ordinary border rank of a symmetric tensor satisfy an inequality analogous to the one for exact rank,

and in this relaxed setting equality is known to hold considerably more often, since border rank computations are governed by the geometry of secant varieties of the Veronese variety and the corresponding secant varieties of the Segre variety, and these two families of secant varieties tend to agree in dimension over broad ranges of order and rank.

Geometric Interpretation via Secant Varieties

Veronese and Segre Pictures

The symmetric rank of T equals, generically, the minimal number of points on the Veronese variety whose linear span contains the point corresponding to T. The ordinary rank equals the analogous quantity for the Segre variety, which parametrizes rank-one tensors without the symmetry constraint. Since the Veronese variety embeds into the space underlying the Segre variety's ambient projective space when tensors are viewed abstractly, the comparison between the two ranks becomes a comparison between secant varieties of these two projective varieties, and the Rank Relation records the precise containment and dimension count statements that follow from this embedding.

Dimension Counts and Expected Rank

The expected symmetric rank of a generic symmetric tensor of order d in n variables is governed by a dimension count balancing the dimension of the space of symmetric tensors against the dimension of the variety of tensors of rank at most r; this is formalized by the Alexander-Hirschowitz theorem, which classifies the finite list of exceptional cases where the naive dimension count fails. The Rank Relation draws on this classification to identify precisely the exceptional loci where the ordinary and symmetric ranks can diverge, since the known counterexamples to Comon's Conjecture occur in the neighborhood of these same exceptional cases.

Consequences for Applications

Decomposition Algorithms

Numerical and symbolic algorithms for computing tensor decompositions, such as generalizations of the Jennrich or simultaneous diagonalization methods and homotopy continuation methods, must take a position on whether to search for the minimal decomposition within the symmetric class or within the full class of rank-one tensors. The Rank Relation determines when it is safe, from the point of view of achieving the true minimal rank, to relax a symmetric decomposition problem into an ordinary tensor decomposition problem, which is often computationally more tractable.

Complexity-Theoretic Consequences

Because the ordinary tensor rank underlies complexity measures for bilinear and multilinear computation, and symmetric tensors arise naturally in the study of quadratic and higher-degree forms, the Rank Relation constrains how complexity lower bounds obtained through symmetric-tensor techniques transfer to bounds on the complexity of general tensors, and vice versa.