15.14.3 Tensor Symmetric Decomposition Coefficient Role

Tensor Symmetric Decomposition Coefficient Role is the function performed by the scalar multipliers c_1 through c_r attached to each pure power term in a symmetric tensor decomposition, specifically their role in scaling the magnitude of each term, absorbing the scalar ambiguity inherent to a single pure power term, and serving as the linear, and therefore comparatively tractable, unknowns within the otherwise nonlinear decomposition relation. While the vectors v_1 through v_r in a decomposition determine the directions along which the tensor is being resolved, the coefficients c_k determine how strongly each of those directions contributes, and separating their role from that of the vectors clarifies which part of the decomposition problem is linear and which part is not.

Recognizing the coefficients as a distinct component of the decomposition, rather than folding them notationally into the vectors themselves, is useful precisely because doing so isolates the one part of the decomposition relation that behaves linearly, a distinction with direct consequences for how decompositions are computed and how the scalar ambiguity of individual pure power terms is resolved.

The Coefficients Within the Decomposition Relation

Linearity in the Coefficients

Once the vectors v_1 through v_r in a decomposition are fixed, the decomposition relation:

becomes a linear system in the unknowns c_1 through c_r, since each symmetric power v_k^{odot n} is by that point a fixed, known tensor, and T is expressed as a linear combination of these fixed tensors; solving for the coefficients, given the vectors, is therefore an ordinary linear algebra problem, solvable by standard techniques such as least squares or exact linear solving when the system is consistent.

Contrast With the Nonlinearity in the Vectors

This linearity stands in direct contrast to the situation when the vectors themselves are unknown, since each symmetric power v_k^{odot n} depends polynomially, of degree n, on the components of v_k; the coefficient role is therefore precisely the part of the overall decomposition problem that remains tractable by linear methods even when the harder, nonlinear search for suitable vectors has not yet been resolved.

Absorbing the Scalar Ambiguity of Pure Power Terms

Compensating for Rescaled Vectors

As noted when examining the pure power term individually, replacing a vector v_k by a scalar multiple lambda v_k requires compensating by dividing the coefficient c_k by lambda to the n in order to leave the term c_k v_k^{odot n} unchanged; the coefficient's role in absorbing this compensation is what allows the vectors in a decomposition to be normalized, for instance to unit length, without altering the tensor being represented, simply by adjusting the coefficients accordingly.

Normalization Conventions

A common convention arising from this absorption role is to require each vector v_k to have unit length under some fixed norm, with the coefficient c_k then carrying the entire magnitude information for that term; this convention cleanly separates the decomposition into a directional part, the normalized vectors, and a magnitude part, the coefficients, mirroring the separation between direction and magnitude familiar from ordinary vector decompositions.

Coefficients and Sign Conventions for Even Degree

Sign Absorption When n Is Even

When the rank n is even, negating a vector v_k leaves its symmetric power v_k^{odot n} unchanged, since negative one raised to an even power is one; consequently the coefficient c_k cannot distinguish between v_k and its negation in this case, and any sign ambiguity in the choice of vector direction is entirely absorbed by the vector itself rather than requiring any special handling from the coefficient.

Sign Significance When n Is Odd

When n is odd, negating v_k also negates its symmetric power, so the coefficient and the vector's sign interact directly; a term c_k v_k^{odot n} can equivalently be written as negative c_k times the negated vector's power, meaning the coefficient's sign and the vector's overall orientation are linked and only their combined effect, not either one individually, is determined by the decomposition relation.

Practical Role in Computing a Decomposition

Two-Stage Solution Strategies

Because the coefficients are linear unknowns once the vectors are fixed, many practical approaches to finding a decomposition proceed in two conceptually separate stages: searching for suitable vectors, the genuinely nonlinear part of the problem, and then solving directly for the corresponding coefficients via linear algebra once candidate vectors have been proposed, using the coefficient role's linearity to avoid folding this simpler subproblem into the harder nonlinear search.

Coefficients as a Diagnostic for Decomposition Quality

Given a proposed set of vectors that only approximately satisfies the decomposition relation, solving the resulting linear system for the best-fitting coefficients, typically via least squares, and examining the residual error between the reconstructed tensor and the true target T provides a direct, quantitative diagnostic of how well the chosen vectors are performing, independent of any further adjustment to those vectors themselves.