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Coded Hankel Polynomial Chaos: Spectral Identification of Dominant Polynomial-Chaos Modes

Zhiliang Deng, Xiaomei Yang

arXiv:2608.16126v1stat.MLcs.LG

TL;DR

Selecting dominant polynomial-chaos modes is difficult in high-dimensional bases, so the paper introduces a Hankel-based spectral formulation that encodes modes through phase-coded exponential sums. Experiments support exact and noisy recovery, unknown-order identification, and dominant-mode recovery for a stochastic Darcy problem.

  • Problem

    Identifying a small dominant support and its coefficients in high-dimensional polynomial-chaos expansions remains important for revealing influential variables, orders, and interactions.

  • Method

    CH-PC converts polynomial-chaos coefficients into phase-coded exponential sums whose low-rank Hankel structure and root-of-unity labels recover active multi-indices.

  • Results

    Numerical examples support exact and noisy Legendre recovery, unknown-order identification by phase persistence, and dominant-mode recovery for a stochastic Darcy problem.

  • Takeaways & Limitations

    CH-PC provides a direct spectral route from observed responses to active stochastic multi-indices without iterative searches through candidate basis functions.

  • Takeaways & Limitations

    The scaling advantage from factorized probe evaluation is restricted to tensor-product candidate sets, while approximate sparsity is treated through structured perturbations rather than a separate recovery theory.

Abstract

from arXiv · show

Identification of dominant polynomial-chaos modes is usually formulated as a sparse-regression problem on a sampled multivariate polynomial dictionary. We develop coded Hankel polynomial chaos (CH-PC), a complementary spectral formulation for dominant-mode identification. A finite generating transform converts PCE coefficients into a coefficient-generating polynomial, and evaluation along a geometric phase orbit produces a finite exponential sum. Its model order and spectral nodes are encoded by low-rank Hankel matrices, while coordinate phase shifts attach root-of-unity labels from which the full polynomial multi-indices are recovered. Coordinate-shifted probes are combined as common-node snapshots, and independent phase encodings provide redundant representations when a single spectral encoding is poorly conditioned. For finite observations, population, finite-data, and observed probes are kept distinct: sampling or quadrature error and observation error enter as separate Hankel perturbations, which are then connected to spectral stability, discrete decoding, and phase voting. For tensor-product candidate sets, the generating kernel factorizes into one-dimensional sums and can be evaluated without assembling the full multivariate PCE design matrix. Numerical experiments on sparse Legendre benchmarks and a stochastic Darcy problem illustrate exact recovery, noise stabilization, unknown-order identification by phase persistence, and dominant-mode recovery for a PDE-generated quantity of interest.

1 Introduction

The paper introduces coded Hankel polynomial chaos as a complementary spectral formulation for identifying dominant polynomial-chaos modes. It converts PCE coefficients into finite exponential models whose Hankel structure, phase shifts, and redundant encodings support model-order and multi-index recovery.

  • Motivation: High-dimensional PCEs can contain many candidate basis modes even at modest degree, although responses often depend significantly on only a small subset.This motivates basis selection for dominant-mode identification.
  • Related work: Existing approaches include least angle regression, orthogonal matching pursuit, ℓ1 regularization, compressed sensing, and adaptive greedy strategies.
  • Spectral formulation: The paper develops a complementary spectral viewpoint using Hankel matrix-pencil techniques to recover finite exponential sums, with Hankel rank representing model order and matrix pencils recovering spectral nodes.The principle traces to classical Prony methods and their generalizations.
  • Core construction: A finite generating transform maps PCE coefficients to a coefficient-generating polynomial, while geometric phase-orbit evaluations produce exponential sums for Hankel-rank order identification and matrix-pencil node recovery.Coordinate phase shifts are introduced because one spectral node contains a multi-index only through a scalar phase combination.
  • Finite-data realization: The framework tracks data-based probe perturbations at the joint-Hankel level before propagating them through spectral recovery and discrete multi-index decoding.Phase conditioning depends on both data error and separation of encoded nodes, motivating repeated decoding over independent phases.
  • Contributions: The main contributions combine a PCE-specific generating representation, root-of-unity coordinate encoding, common-node snapshots, and independent phase encodings for redundant spectral recovery.The generating representation yields an exact Hankel characterization of effective model order, while coordinate shifts reconstruct individual degrees and full active multi-indices.

2 Dominant polynomial-chaos modes and exact spectral representation

This section defines dominant polynomial-chaos modes through coefficient energy and develops an exact population-level spectral representation. A finite generating transform, phase-coded exponential sums, and Hankel factorizations enable model-order, support, and coefficient recovery, with coordinate shifts decoding full multi-indices.

  • Generating transform: The finite generating transform converts orthogonal PCE coefficients into an ordinary multivariate coefficient-generating polynomial while preserving complete multi-index support.It requires no derivative of the quantity of interest and is independent of the orthogonal-polynomial family.
  • Dominant modes: Dominant support is defined as a small subset of candidate multi-indices that captures principal coefficient energy, with exact and approximate sparsity distinguished.The identification target includes the support, its effective cardinality, and associated coefficients; multi-indices also encode interaction structure.
  • Computational structure: For tensor-product candidate sets, kernel probes factor into one-dimensional sums, but this exact product factorization need not hold for total-degree or general finite sets.The resulting Hankel and pencil computations depend on the model-order ceiling rather than the full candidate-set cardinality.
  • Spectral representation: Evaluating the transform on a geometric phase orbit produces a finite exponential sum whose unknown order can be recovered from the rank of a sufficiently dimensioned Hankel matrix.With R,C ≥ smax, exact Hankel and reduced-pencil methods recover the model order and spectral nodes when amplitudes are nonzero and nodes are distinct.
  • Multi-index decoding: Coordinate-shifted probes preserve common Prony nodes while applying root-of-unity amplitude codes, reducing multi-index recovery to coordinate-wise discrete decoding.For retained coordinate ℓ, γℓ = 2π/(pℓ + 1) bijectively maps admissible degrees to (pℓ+1)-st roots of unity.
  • Exact recovery: Under nonzero amplitudes and pairwise distinct nodes, the Hankel rank, reduced pencil, and coordinate codes uniquely recover model order, spectral nodes, active multi-indices, and PCE coefficients.Distinct-node recovery holds almost surely for a random absolutely continuous phase choice.

3 Noise-robust coded Hankel recovery

This section develops a finite-data CH-PC decoder by jointly recovering common spectral nodes from coordinate-shifted Hankel snapshots. It separates sampling, observation, and phase-conditioning effects, then uses stability conditions, discrete decoding, and phase voting for robust support recovery.

  • Common-node joint Hankel recovery: Coordinate-shifted probes share the base sequence’s Prony nodes, so stacking all d+1 Hankel snapshots jointly estimates the common spectral subspace.The subsequent coordinate-decoding rule remains unchanged; only the spectral estimation step is modified.
  • Coefficient estimation: Support identification is separated from coefficient estimation, avoiding the use of noisy Vandermonde amplitudes as final coefficient estimates.After common nodes are recovered, base and coordinate-shifted probes are fit separately and shifted-to-base amplitude ratios decode coordinates.
  • Perturbation decomposition: Finite-data sampling or quadrature error and observation noise enter as separate Hankel perturbations, with exact probe covariance determined by the basis, phases, inputs, radii, and weights.The total fixed-phase radius is bounded by ηH(θ) + ΔH(θ, δ), while exact quadrature makes ηH(θ) = 0.
  • Model-order identification: If ||E||2 ≤ Δ and σ_s(J0) > 2Δ, thresholding singular values at Δ recovers the exact model order s.The perturbed matrix retains σ_s(Ĵ0) > Δ while σ_{s+1}(Ĵ0) ≤ Δ.
  • Stability and phase voting: Within a phase-dependent stability radius, the joint decoder recovers the exact PCE support under exact sparsity, while random phase voting supplies redundant recovery across independently drawn encodings.The phase-voting results distinguish finite-data error, observation noise, conditional spectral stability, and redundancy from the number of phases Q.

4 Approximate sparsity, sensitivity, and methodological scope

Approximate sparsity appears as a third structured perturbation in CH-PC, so recovered modes are interpreted as resolvable dominant support rather than provably nonzero-only coefficients. The section also bounds the method’s scope through phase–radius design, noise and integration assumptions, and its role as a spectral framework rather than a universal replacement for sparse regression.

  • Approximate sparsity: Approximate sparsity enters CH-PC as a third structured perturbation alongside finite-data integration error and observation noise.Weak nonzero coefficients are incorporated without a separate recovery theory, through an additional structured Hankel perturbation.
  • Approximate sparsity: CH-PC identifies modes resolvable relative to dominant amplitudes, radius attenuation, weak-tail energy, data error, observation noise, and spectral conditioning—not coefficients proved exactly zero.This yields a resolution-based interpretation of dominant support consistent with coefficient-energy sensitivity measures.
  • Sensitivity: Recovered multi-indices identify participating variables and their corresponding contribution to the variance decomposition.For an orthonormal PCE, nonconstant coefficient energy equals response variance, linking mode identification to Sobol interpretation.
  • Methodological scope: Optimal joint phase–radius design remains open because phase separation must be balanced against probe amplification and attenuation of high-order modes.The proposed covariance-based diagnostic captures only part of this tradeoff.
  • Methodological scope: The framework extends beyond Gaussian observation error to independent sub-Gaussian and heteroscedastic noise, while heavy-tailed contamination requires robust probe-functional estimators.Heteroscedastic noise can be represented using a known or estimated observation covariance.
  • Methodological scope: CH-PC is a spectral support-identification framework, not a universal replacement for sparse regression.The methods have different conditioning mechanisms, so the numerical study emphasizes recovery behavior, joint snapshots, and phase redundancy without fixing a method ordering.

5 Numerical Experiments

The numerical experiments test CH-PC on a shared sparse Legendre benchmark and a stochastic Darcy quantity of interest, covering exact recovery, noise robustness, unknown-order identification, and dominant-mode recovery. Results show exact finite-rank recovery, stabilization from joint snapshots and phase voting, persistence-based order estimation, and successful recovery of four dominant PDE modes.

  • Exact recovery: The noiseless decoder recovers the four-mode support up to permutation, with a sharp rank-four Hankel structure and singular values below 10^-13 after the first four.The first four singular values lie approximately between 10^1 and 10^-1, confirming the finite-rank mechanism.
  • Unknown-order recovery: With unknown order and Q = 13, exact support-and-order frequency is 0.950 at 20% noise and 0.735 at 30% noise, while mean recall remains 0.9388 at 30%.At 30% noise, increasing phase encodings raises frequency from 0.444 for Q = 3 to 0.900 for Q = 17, with mean selected order moving from 3.839 to 3.978.
  • Stochastic Darcy problem: For the stochastic Darcy quantity of interest, CH-PC, LARS, and OMP recover all four reference dominant modes through 20% noise, with median relative errors below 1.6 × 10^-2.The reference support is defined by the four largest nonconstant coefficients from a higher-accuracy projection.

6 Conclusion

CH-PC formulates dominant polynomial-chaos mode identification as a spectral problem: generating transforms and geometric phase sampling encode PCE modes in low-rank Hankel matrices, while coordinate shifts recover discrete multi-index labels. The conclusion highlights separate perturbation tracking, factorized tensor-product computation, numerical validation, and open theoretical questions.

  • Spectral formulation: A finite generating transform and geometric phase sampling convert PCE coefficients into exponential sums whose model order and nodes are encoded by low-rank Hankel matrices.Coordinate shifts preserve Prony nodes and attach finite root-of-unity labels, enabling coordinate-wise integer recovery.
  • Perturbation and redundancy: Distinct population, finite-data, and observed probes separate integration and observation errors at the joint-Hankel level.Joint coordinate snapshots reuse common spectral nodes, while independent random phase maps provide redundant encodings for phase separation and majority voting.
  • Computational structure: For tensor-product candidate sets, the generating kernel factorizes into one-dimensional polynomial sums, avoiding assembly of the full multivariate PCE design matrix.The resulting complexity differs from dictionary-based sparse regression, but total cost depends on the number of phase decoders and no universal runtime ordering is claimed.
  • Numerical evidence: Numerical examples demonstrate exact and noisy Legendre recovery, unknown-order identification by phase persistence, and dominant-mode recovery for a stochastic Darcy quantity of interest.In the Darcy problem, the dominant PCE modes are induced by the PDE response rather than prescribed in advance.
  • Open questions: Future theory targets sharper sample-complexity bounds, noise-aware optimization of radii and phases, and structured probe evaluation.The proposed bounds are to depend on observation count, active amplitude, phase separation, and code redundancy.

A A square-pencil perturbation estimate

This appendix states the square-pencil perturbation estimate supporting the local stability argument. It also gives the root-of-unity separation used to guarantee exact nearest-root decoding.

  • A square-pencil perturbation estimate: The appendix records a standard reduced-pencil perturbation estimate for the local stability argument, separate from the PCE-specific main text.It is presented as supporting linear algebra rather than a PCE-specific ingredient.
  • A square-pencil perturbation estimate: For diagonalizable T = XZX^-1, every eigenvalue of the perturbed pencil lies within κ2(X)^2 of the exact spectrum.The bound is identified as the Bauer–Fike estimate.
  • A square-pencil perturbation estimate: For coordinate ℓ, adjacent root-of-unity codes are separated by d_ℓ = 2 sin(π/(p_ℓ + 1)), enabling exact nearest-root decoding under the stated perturbation condition.The supplied passage gives the condition in terms of the perturbed code distance, but its inequality is truncated.

B Regression baselines

LARS and OMP serve as established reference methods for numerical comparisons with CH-PC. The comparisons use identical evaluations, points, weights, candidate sets, and, in oracle-order experiments, supplied true model order.

  • B Regression baselines: LARS and OMP are treated as established reference methods alongside CH-PC.
  • B Regression baselines: All three methods receive exactly the same model evaluations, input points, weights, and finite PCE candidate set.For candidate set A, the weighted regression matrix has entries Φn,α = √wnΨα(ξ(n)).
  • B Regression baselines: In oracle-order experiments, the true number of retained modes is supplied to all three methods, isolating support identification from model-order selection.
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