15.14.1 Tensor Symmetric Decomposition Term Set

Tensor Symmetric Decomposition Term Set is the collection of vectors, and the multiset of the rank-one symmetric tensors built from them, that together form a minimal (or otherwise specified) symmetric decomposition of a given symmetric tensor. Where the symmetric rank is a number, the term set is the actual data realizing a decomposition attaining that number, and the properties of this set, rather than merely its size, govern uniqueness, identifiability, and the sensitivity of the decomposition to perturbation.

Definition and Basic Objects

The Decomposition Data

Given a symmetric tensor T of order d over a vector space V, a symmetric decomposition of size r consists of an ordered or unordered collection of r vectors, together with the requirement that the sum of their d-th tensor powers reproduces T:

The Term Set is precisely the set (or, when repetitions or coincidences are permitted, the multiset) of vectors v_1 through v_r appearing in such an expression, considered up to the natural equivalences of the problem: reordering of the terms, and rescaling of each vector by a d-th root of unity, since replacing v_i by a scalar multiple that is a d-th root of unity leaves the d-th tensor power unchanged.

Projectivization

Because only the d-th tensor power of each vector matters, the term set is more naturally recorded as a set of points in the associated projective space, each point carrying a scalar coefficient absorbing the magnitude and any remaining root-of-unity ambiguity. This projective packaging identifies the Term Set with a finite set of points lying on the Veronese variety, together with coefficients, and this is the object directly produced or certified by algorithms based on apolarity and secant variety geometry.

Existence and Cardinality

Minimal versus Non-Minimal Term Sets

A term set realizing the symmetric rank of T is called a minimal term set; term sets of larger cardinality that still sum to T are non-minimal and typically exist in abundance once at least one exists, since padding a minimal decomposition with cancelling pairs of terms (terms that sum to zero after being added and subtracted) produces valid, larger decompositions. The Term Set concept is most informative when restricted to minimal decompositions, or to decompositions of a specified target size relevant to an application, such as a bounded model complexity in a statistical or signal-processing setting.

Existence for Every Size Above the Rank

If a symmetric tensor admits a term set of size r, it also admits term sets of every size strictly greater than r, up to the dimension considerations of the ambient space, since additional cancelling or redundant terms can always be appended. The nontrivial existence question is therefore concentrated at and near the minimal size, where the geometry of secant varieties of the Veronese variety determines whether a term set of that size exists at all.

Uniqueness of the Term Set

Generic Uniqueness

For generic symmetric tensors of a given order and in a given number of variables, when the symmetric rank is below a threshold governed by the Alexander-Hirschowitz theorem and refined identifiability results, the minimal term set is unique. This uniqueness is the basis for calling such decompositions identifiable, meaning that the term set can, in principle, be recovered exactly from the tensor itself, without ambiguity beyond the trivial rescaling equivalences already accounted for.

Non-Uniqueness and Positive-Dimensional Families

Above the identifiability threshold, or in special exceptional strata classified by the Alexander-Hirschowitz theorem, a symmetric tensor may admit infinitely many minimal term sets, forming a positive-dimensional family. The classical example is the case of binary quadratic and certain binary quartic forms, where the fibers of the decomposition problem are described explicitly by classical invariant theory, and the general phenomenon is studied through the geometry of secant varieties and their fibers over points of high multiplicity secant defect.

The Kruskal Uniqueness Criterion

In the closely related non-symmetric decomposition setting, and adapted to the symmetric case, a term set can be certified unique using a rank-based criterion on the vectors and their sub-collections: if every proper subset of the term set spans a subspace of the expected dimension (a condition phrased in terms of the Kruskal rank of the associated matrix of vectors), then the decomposition is the unique one of that size. This criterion supplies a checkable, purely linear-algebraic sufficient condition for term set uniqueness that does not require exhaustive search.

Computing the Term Set

Apolarity-Based Recovery

Because the Term Set corresponds to a set of points on the Veronese variety, and these points are cut out by the apolar ideal of T, the term set can in principle be recovered by finding the points of an apolar set, i.e., a set of reduced points whose ideal is contained inside the apolar ideal of T and whose cardinality matches the rank. Catalecticant matrices and their kernels supply the linear algebra needed to write down candidate apolar ideals, and, in low-rank cases, the corresponding points can be extracted symbolically.

Numerical and Algorithmic Extraction

In practice, for numerical tensors, the term set is extracted by numerical linear algebra techniques generalizing eigenvalue decomposition, or by direct optimization that minimizes the discrepancy between T and a candidate sum of d-th powers over a chosen number of terms, seeded from multiple random initializations to guard against convergence to non-global critical points. Homotopy continuation methods, tracking solution paths from a related tensor with a known term set to the target tensor, provide an alternative, certifiably complete method of enumeration.

Sensitivity and Conditioning

Perturbation of the Term Set

Even when the Term Set is combinatorially unique, its numerical conditioning can vary widely: term sets containing nearly collinear or nearly coincident vectors correspond to ill-conditioned decompositions, in which small perturbations of T induce disproportionately large changes in the recovered vectors. This conditioning is quantified through the smallest singular value of derivative maps associated with the decomposition, and it governs the practical reliability of any numerically recovered term set.

Role in Applications

In applications such as blind source separation, higher-order statistics, and algebraic statistics, the Term Set is the object of direct scientific interest, since its individual vectors are interpreted as latent sources, mixture components, or parameters of a model, and the symmetric rank alone, without the accompanying term set, does not answer the underlying scientific question.