Polynomial Spectral Iterations

• Polynomial spectral iterations are techniques that utilize algebraic and spectral properties of polynomials and operators to enhance optimization and filtering tasks.
• They integrate methods like hierarchical spectral relaxations, iterative transforms, and orthogonal polynomial modifications to achieve rapid convergence and scalability.
• Applications span operator theory, graph signal processing, and nonlinear functional evaluations, offering efficient alternatives to traditional sum-of-squares and SDP methods.

Polynomial spectral iterations encompass a suite of methodologies that leverage the algebraic and spectral properties of polynomials, operators, and associated matrices to enable efficient optimization, functional transformation, filtering, or factorization in structured algebraic or geometric settings. They unify themes from operator theory, polynomial optimization, graph spectra, orthogonal polynomials, and numerical algorithms by systematically exploiting polynomial mapping, spectral decompositions, and iterative application of polynomial transforms.

1. Hierarchical Spectral Relaxations for Polynomial Optimization

Polynomial optimization over varieties defined by polynomial equations can be approached via a spectral-relaxation hierarchy. Given a minimization problem,

$f^* = \min_{x \in V(I)} p(x),$

where $V(I)$ is an algebraic variety defined by polynomial constraints, the relaxation is constructed by seeking a scalar $\gamma$ such that $p(x) - \gamma \geq 0$ for all $x \in V(I)$. In the quotient ring $\mathbb{R}[x]/I$, this condition is translated to the sum-of-squares representation: $$p - \gamma \equiv z(x)^T(M(p) - \gamma M(1))z(x)\ (\mathrm{mod}\ I),$$ where $z(x)$ is a lifted monomial vector, and $M(1) \succ 0$, $M(p)$ are symmetric Gram matrices. The maximal feasible $V(I)$0 coincides with the smallest generalized eigenvalue: $V(I)$1 This process is generalized through a graded hierarchy using collections of I-spherical polynomials $V(I)$2 satisfying $V(I)$3. At level $V(I)$4, basis vectors span

$V(I)$5

yielding Gram matrices $V(I)$6, $V(I)$7. These are constructed and updated via least-squares projections and Kronecker-type lifts, and at each level, the smallest generalized eigenvalue $V(I)$8 forms an ascending chain of lower bounds converging to $V(I)$9: $\gamma$0 The computational architecture requires only eigenvalue computations on stably sized matrices, circumventing the scalability bottleneck of SDP-based sum-of-squares (SOS) relaxations. Numerical experiments validate dramatic speedups while yielding bounds close to those from traditional SOS (Moreno et al., 22 Jul 2025).

2. Polynomial Spectral Iterations in Operator Theory

In operator-theoretic settings, polynomial spectral iterations refer to transfinite or ordinal-indexed iterative application of polynomial (or holomorphic) transforms to bounded operators, particularly in Hilbert or Banach spaces. For a normal operator $\gamma$1 on Hilbert space and a contractive polynomial transform $\gamma$2 (i.e., $\gamma$3), the iteration: $\gamma$4 stabilizes at countable ordinal $\gamma$5. The limit $\gamma$6 is the orthogonal projection onto the peripheral fixed-point set: $\gamma$7 This convergence is governed by spectral mapping theorems and Fejér-type monotonicity arguments. In Banach-space settings (Ritt and sectorial operators), the mean-ergodic projection of iterated holomorphic polynomial transforms yields an idempotent commuting with the original operator, providing ergodic-type decompositions under bounded $\gamma$8 functional calculus (Alpay et al., 8 Aug 2025).

3. Iterative Spectral Transformations of Orthogonal Polynomials

Polynomial spectral iterations also encompass sequences of Christoffel, Geronimus, and Uvarov transformations on orthogonal polynomials. In the multivariate setting, each transformation modifies a linear functional $\gamma$9 on $p(x) - \gamma \geq 0$0, and hence the associated moment (Hankel) matrix and monic multivariate orthogonal polynomials (MVOP). Each Christoffel or Geronimus step is realized through operator transformations and linked by resolvents: $p(x) - \gamma \geq 0$1 with subsequent composition corresponding to higher-order Darboux transformations. Explicit closed-form formulas via quasi-determinants express the result of multiple spectral iterations on MVOP and the quasi-tau norms. In integrable systems, iteration of these transforms encodes the evolution of Lax matrices under the multispectral 2D Toda hierarchy, with spectral iterations thus providing the algebraic backbone for families of integrable hierarchies (Ariznabarreta et al., 2015).

4. Polynomial Spectral Filtering on Graphs and Networks

Polynomial spectral filtering constitutes a central technique in graph signal processing and distributed computation. A polynomial filter $p(x) - \gamma \geq 0$2 of the graph Laplacian $p(x) - \gamma \geq 0$3 can be used for low-pass filtering, consensus acceleration, or gossip algorithms—either as a fixed-degree polynomial (e.g., Chebyshev, Jacobi) or via adaptive Krylov-based iteration. The action: $p(x) - \gamma \geq 0$4 can be implemented without explicit diagonalization by leveraging Krylov subspace projections and three-term recurrences.

• In average consensus, optimally designed second-order polynomials can achieve finite-time convergence if the graph spectrum clusters appropriately, with worst-case contraction rates:

$p(x) - \gamma \geq 0$5

where $p(x) - \gamma \geq 0$6, $p(x) - \gamma \geq 0$7 are spectral extremal points (Apers et al., 2015).

• In spectral graph denoising, Krylov subspace methods such as conjugate gradient and LOBPCG adaptively construct polynomial filters yielding rapid convergence relative to iterative local filtering (Knyazev et al., 2015).
• Jacobi polynomial-based gossip leverages an approximation of the graph's spectral measure by Jacobi measures, aligning the polynomial iteration with the spectral dimension for optimal transient convergence rates in high-dimensional lattices or percolation graphs (Berthier et al., 2018).

5. Sparse Spectral Iterations for Nonlinear Functionals

In numerical spectral computation, the evaluation of polynomial or general nonlinear functionals $p(x) - \gamma \geq 0$8 in a spectral basis is accelerated by restriction to a sparse index set determined by the decay properties of the coefficients. For smooth functions, the effective computation of each coefficient $p(x) - \gamma \geq 0$9 is restricted to tuples $x \in V(I)$0 satisfying $x \in V(I)$1, reducing complexity from $x \in V(I)$2 for naive summation to $x \in V(I)$3 or $x \in V(I)$4 for the iterative pairwise algorithm. Convergence rates are algebraic and governed by the regularity of the input and the decay of structure coefficients, with explicit error bounds in weighted $x \in V(I)$5 spaces (Faou et al., 2012).

6. Polynomial-Spectral Factorizations in Graph Composition

Advanced graph-theoretic constructions exploit iterative polynomial-spectral factorizations arising from generalized graph composition (H-join) operads. Given a composition encoded by a Schröder tree, at each level, the adjacency and Laplacian spectra of the composite graph factor into small block matrix determinants depending on constituent regularities and join structure. The factorization theorem yields characteristic polynomial decompositions: $x \in V(I)$6 where $x \in V(I)$7 are inter-block matrices at internal nodes. The same paradigm extends to Laplacian, generalized, and universal adjacency polynomials, providing a systematic operad-theoretic framework for recursive spectral analysis and factorization over arbitrary levels of graph composition (Liendo, 29 Apr 2026).

7. Convergence and Stability Considerations

Across domains, the convergence and stability of polynomial spectral iterations are dictated by spectral gap, peripheral spectrum structure, and contractivity conditions on the applied polynomial transforms.

• In operator contexts, stabilization is provably achieved at countable stage $x \in V(I)$8, with the limit being an orthogonal or mean-ergodic projection.
• In distributed systems, acceleration schemes leveraging higher-order polynomials can exhibit fragility to changes in network topology (e.g., edge failures), necessitating constraints on permissible polynomials or preconditioning strategies to ensure per-step contractivity (Apers et al., 2015).
• In stochastic eigenvalue problems, spectral inverse iteration demonstrates geometric convergence depending on the maximum second-to-first eigenvalue ratio and polynomial approximation regularity (Hakula et al., 2017).

The spectral-iteration framework thus provides a unifying algebraic and computational formalism applicable in polynomial optimization, operator theory, integrable systems, graph algorithms, and numerical analysis, characterized by explicit control over convergence, algebraic structure, and scalability.