Polynomial Coefficient Decomposition

• PCD is a framework that represents polynomial maps as structured sums over lower-complexity components, such as rank-one terms or powers of linear forms.
• It provides efficient methods for decomposition, identifiability analysis, and rank bounding, leveraging tools like catalecticants and secant variety theory.
• PCD underpins applications in signal processing, machine learning, and neural architectures by enabling scalable recovery of polynomial and tensor structures.

Polynomial Coefficient Decomposition (PCD) is a unifying framework for expressing polynomials and polynomial maps as structured sums over lower-complexity building blocks, typically rank-one terms or powers of linear forms. PCD generalizes classical tensor decompositions (e.g., CP, symmetric/Waring), algorithmic approaches in polynomial algebra, and recent applications in signal processing, machine learning, algebraic geometry, and graph neural networks. It rigorously addresses identifiability, generic and maximal rank, algorithmic recovery, and parameter efficiency for a broad class of models, with particular relevance to the decomposition of polynomial maps, the analysis of secant varieties, and multilinear architectures.

1. Formal Definition and Decoupled PCD Model

A polynomial map $f: K^m \rightarrow K^n$ (where $K = \mathbb{R}$ or $\mathbb{C}$, $d \geq 1$) admits an $r$-term decoupled Polynomial Coefficient Decomposition if there exist vectors $v_1,\dots,v_r \in K^m$, $w_1,\dots,w_r \in K^n$ and univariate polynomials $g_k(t) = \sum_{\ell=1}^d c_{\ell,k} t^\ell$ such that

$f(u) = \sum_{k=1}^r w_k g_k(v_k^\top u).$

Equivalently, in matrix form: $f(u) = W\; g(V^\top u),$ where $V \in K^{m \times r}$ and $W \in K^{n \times r}$ (Comon et al., 2016).

Specializations of the PCD model encompass:

• Matrix factorization ($d=1$),
• Ridge polynomial sums for scalar output ($n=1$),
• Simultaneous Waring decompositions for homogeneous $f$ of degree $d$.

2. Algebraic Geometry: X-Rank and Veronese Scroll

PCD extends tensor decompositions by interpreting the decomposed structures via X-rank on projective varieties. Let $A \simeq K^N$, and $\hat{X} \subset A$ be a scale-invariant, irreducible, nondegenerate algebraic cone. Given $v \in A$, its X-rank is

$$\mathrm{rank}_X(v) = \min\{ r : v = x_1 + \cdots + x_r,\ x_k \in \hat{X} \}.$$

The minimal such expression is an X-rank decomposition. Critical to PCD is the Veronese scroll variety $\hat{X}_{a,V}$, defined as the image of

$$\psi(v, (c_1, \dots, c_d)) = (c_1 v^{\otimes a_1},\dots, c_d v^{\otimes a_d}),$$

typically with $a=(1,\dots,d)$. For the full PCD model, one considers the Segre product $$\widehat{Y} = \mathrm{Seg}(\hat{X}_{(1,\dots,d),V} \times W)$$, and studies secant varieties $\sigma_r(\widehat{Y})$ to analyze rank and identifiability (Comon et al., 2016).

3. Rank Bounds and Identifiability

The generic and maximal rank for PCD admits tight, dimension-dependent bounds. For $d \geq 4$, $m > (d-1)^2$,

$$r_{\text{gen}}(\widehat{Y}) = \left\lceil \frac{\binom{m+d-2}{d-1} + \binom{m+d-1}{d}/(m+1)}{1} \right\rceil,$$

with maximal rank

$$r_{\text{max}}(\widehat{Y}) \leq 2\, r_{\text{gen}}(\widehat{Y}).$$

For scalar-output ridge-sum ($n=1$), this improves the classical symmetric tensor bound $r_{\text{max}} \leq \binom{m+d-2}{d-1}$.

Identifiability is established as follows: $\widehat{Y}$ is $r$-identifiable if, for general $y \in \sigma_r(\widehat{Y})$, the decomposition is unique up to permutation. Sufficient identifiability holds for

$r \leq \min(\lceil r_5(m,n,d)\rceil-1, m)\cdot n,$

with $r_5(m,n,d)$ the minimum of the dimension-based weak-defectivity bound and the scroll dimension (Comon et al., 2016).

Partial identifiability is guaranteed for coefficients of degree $\geq s$, provided $r$ does not exceed $\binom{m+s-1}{s} n$.

4. Computational Methods for Decomposition

Recovery of the PCD involves explicit algebraic constructions:

• Linearization via stacked catalecticant matrices whose $2\times2$ minors characterize the Veronese scroll,
• Solving for $v_k, w_k$ by projection onto top-degree slices and simultaneous Waring decomposition leveraging classic identifiability results for Segre-Veronese varieties,
• Subsequent recovery of lower-degree coefficients $c_{\ell,k}$ by linear systems conditioned on full-rank catalecticant blocks (Comon et al., 2016).

For homogeneous polynomials or symmetric tensors, classical algorithms include Hankel moment matrices, apolar ideals, quotient algebras, multiplication operator simultaneous diagonalization, and solving for weights by coefficient matching (Laface et al., 2021). The Sylvester/catalecticant, Koszul flattening, and bundle methods provide practical procedures, with computable bounds on rank and uniqueness (Oeding et al., 2011).

For non-homogeneous power-sums of degree-$K$ polynomials $P(x) = \sum_{t=1}^m A_t(x)^d$, span extraction via partial derivatives and "desymmetrization" enables polynomial complexity algorithms that are stable to inverse-polynomial noise and work with generic $A_t(x)$ up to $m = \tilde{O}(n)$ when $K=2, d=3$ (Bafna et al., 2022).

5. Applications in Learning and Signal Processing

PCD underpins architectural design in spectral graph convolutional networks (SGCNs), where spectral filters are polynomials of a graph Laplacian or adjacency. The coefficient tensor $$\mathcal{W} \in \mathbb{R}^{d_{\text{in}}\times d_{\text{out}}\times K}$$ admits classical CP and Tucker decompositions: $$\mathcal{W} \approx \sum_{r=1}^R c_r \circ p_r \circ m_r\quad \text{(CP)},$$

$$\mathcal{W} \approx \mathcal{G} \times_1 A \times_2 B \times_3 C\quad \text{(Tucker)},$$

yielding SGCN layers with significant parameter reductions and controlled expressivity. Empirically, CoDeSGC-CP and -Tucker outperform baseline polynomial convolutions on classification tasks, particularly in settings with heterophilic graphs and medium-sized datasets (Huang et al., 2024).

6. Integer Decomposition and Bitwise Factorization

The "binary method of integer decomposition" formulates factorization of odd integers $M$ as a search for polynomials $p(x), q(x)$ with binary coefficients, such that $M = p(2)q(2)$. Matching coefficients leads to systems of quadratic (and after reduction, linear) equations, solved deterministically via bitwise carry analysis. The worst-case cost is $O(\sqrt{M}(\log_2 M)^4)$, and while not competitive with subexponential factoring, this method illustrates the algebraic generality of PCD in discrete settings (Gao, 2018).

7. Algebraic-Geometric Tools and Theoretical Techniques

Key concepts in the theoretical analysis of PCD include:

• Secant varieties $\sigma_r(\hat{X})$ and their dimension/defectivity properties,
• The Alexander-Hirschowitz theorem for the generic rank of symmetric tensors,
• Weak-defectivity and Terracini’s lemma to establish identifiability bounds,
• Koszul flattenings, catalecticants, and inheritance properties of ideals in projective geometry,
• Eigenvector methods for rank-one extraction in symmetric tensors (Comon et al., 2016, Oeding et al., 2011).

These techniques equip PCD with sharp criteria for uniqueness, robust dimension-dependent rank bounds, and effective algebraic/numerical recovery mechanisms.

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A plausible implication is that PCD serves as a common language for structured decompositions across polynomial algebra, tensor methods, algebraic geometry, numerical algorithms, and neural architectures. It continues to motivate both foundational research (e.g., in secant variety theory, identifiability conditions, and stability analysis) and algorithmic innovation targeting scalable and interpretable models in data science.