[Paper Review] A General Framework for Enhancing Sparsity of Generalized Polynomial Chaos Expansions

Xiu Yang, Xiaoliang Wan+2 more|arXiv (Cornell University)|Jul 10, 2017

Probabilistic and Robust Engineering Design4 citations

TL;DR

This paper proposes a general framework to enhance sparsity in generalized polynomial chaos (gPC) expansions using iterative rotational transforms of random variables via an alternating direction method. By identifying optimal rotation matrices that minimize mutual coherence of the measurement matrix, the method improves compressive sensing accuracy and efficiency, especially for non-Gaussian distributions and high-dimensional problems (up to O(100)), with demonstrated success in Legendre and Chebyshev expansions and as a generalization of prior Hermite-based methods.

ABSTRACT

Compressive sensing has become a powerful addition to uncertainty quantification when only limited data is available. In this paper we provide a general framework to enhance the sparsity of the representation of uncertainty in the form of generalized polynomial chaos expansion. We use alternating direction method to identify new sets of random variables through iterative rotations such that the new representation of the uncertainty is sparser. Consequently, we increases both the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. We demonstrate that the previously developed iterative method to enhance the sparsity of Hermite polynomial expansion is a special case of this general framework. Moreover, we use Legendre and Chebyshev polynomials expansions to demonstrate the effectiveness of this method with applications in solving stochastic partial differential equations and high-dimensional (O(100)) problems.

Motivation & Objective

• To address the challenge of constructing sparse gPC expansions when only limited samples are available in uncertainty quantification (UQ).
• To extend rotation-based sparsity enhancement beyond Hermite polynomials to general gPC expansions with arbitrary probability distributions.
• To improve the accuracy and efficiency of compressive sensing-based UQ by reducing mutual coherence of the measurement matrix through optimal variable rotation.
• To provide a unified framework that generalizes prior rotation-based methods, including iterative rotation for Hermite polynomials.
• To enable effective UQ for complex systems with high-dimensional or non-Gaussian inputs using fewer simulations or experiments.

Proposed method

• An alternating direction method is used to iteratively identify rotation matrices that transform the original random variables into a new set with sparser gPC representation.
• The rotation is determined by maximizing the variation of the quantity of interest (QoI) through singular value decomposition (SVD) of gradients across parameter space.
• The method optimizes the rotation matrix to minimize the mutual coherence of the measurement matrix, enhancing compressive sensing performance.
• The framework is applicable to any gPC basis (e.g., Legendre, Chebyshev, Hermite) and does not require Gaussian assumptions on input variables.
• The approach integrates with standard compressive sensing solvers such as ℓ₁ minimization, OMP, and ℓ₁-2 minimization.
• The method is extended to potentially non-orthogonal transformations in future work, allowing for even greater sparsity and accuracy.

Experimental results

Research questions

• RQ1 Can a general framework be developed to enhance sparsity in gPC expansions beyond Hermite polynomials, especially for non-Gaussian distributions?
• RQ2 How does iterative rotation of random variables affect the mutual coherence of the measurement matrix in compressive sensing-based UQ?
• RQ3 To what extent does the proposed method reduce the number of required samples for accurate gPC reconstruction in high-dimensional problems?
• RQ4 Is the iterative rotation method for Hermite polynomials a special case of this general framework, and how does it compare in performance?
• RQ5 What are the limitations of the method when applied to non-symmetric or degenerate distributions such as Laguerre polynomials?

Key findings

• The proposed framework generalizes prior rotation-based methods, with the iterative Hermite polynomial method [56] being a special case.
• For Legendre and Chebyshev expansions, the method significantly improves sparsity and reduces mutual coherence, as shown by increased μ from 0.15 to 0.45 and 0.50, respectively.
• In high-dimensional problems (d=12, N=455, M=180), the method maintains or improves accuracy with fewer samples compared to standard compressive sensing.
• The mutual coherence μ increases after rotation for Legendre (0.15 → 0.45) and Chebyshev (0.15 → 0.50), indicating better conditioning for ℓ₁ minimization.
• The method is less effective for non-symmetric distributions like Laguerre polynomials, where numerical tests show failure in some cases due to measurement matrix degeneration.
• The framework enables accurate surrogate modeling with reduced computational cost, especially beneficial for expensive simulations or experiments.