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

arXiv: https://arxiv.org/abs/2608.16126 · HTML: https://mlsanity.com/paper/2608.16126/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 21 references that are in the map.
- First posted on arXiv 17 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark](https://mlsanity.com/paper/2002.01290/) (2020)
- [Stability and Super-resolution of MUSIC and ESPRIT for Multi-snapshot Spectral Estimation](https://mlsanity.com/paper/2105.14304/) (2021)
- [Mathematical Foundation of Sparsity-based Multi-snapshot Spectral Estimation](https://mlsanity.com/paper/2202.11189/) (2022)
- [The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution](https://mlsanity.com/paper/2404.03885/) (2024)

## Abstract (arXiv)

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.