# Sparse Bayesian Learning for Complex-Valued Rational Approximations

arXiv: https://arxiv.org/abs/2206.02523 · HTML: https://mlsanity.com/paper/2206.02523/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 53 references that are in the map.
- First posted on arXiv 6 Jun 2022.

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

- [Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark](https://mlsanity.com/paper/2002.01290/) (2020)
- [Compressive sensing adaptation for polynomial chaos expansions](https://mlsanity.com/paper/1801.01961/) (2018)
- [Sparse Polynomial Chaos expansions using Variational Relevance Vector Machines](https://mlsanity.com/paper/1912.11029/) (2019)
- [A Sparse Bayesian Deep Learning Approach for Identification of Cascaded Tanks Benchmark](https://mlsanity.com/paper/1911.06847/) (2019)

## Built on by (most cited first, 2 of 2)

- [Rational kernel-based interpolation for complex-valued frequency response functions](https://mlsanity.com/paper/2307.13484/) (2023)
- [Maximum a Posteriori Estimation for Linear Structural Dynamics Models Using Bayesian Optimization with Rational Polynomial Chaos Expansions](https://mlsanity.com/paper/2408.03569/) (2024)

## Abstract (arXiv)

Surrogate models are used to alleviate the computational burden in engineering tasks, which require the repeated evaluation of computationally demanding models of physical systems, such as the efficient propagation of uncertainties. For models that show a strongly non-linear dependence on their input parameters, standard surrogate techniques, such as polynomial chaos expansion, are not sufficient to obtain an accurate representation of the original model response. Through applying a rational approximation instead, the approximation error can be efficiently reduced for models whose non-linearity is accurately described through a rational function. Specifically, our aim is to approximate complex-valued models. A common approach to obtain the coefficients in the surrogate is to minimize the sample-based error between model and surrogate in the least-square sense. In order to obtain an accurate representation of the original model and to avoid overfitting, the sample set has be two to three times the number of polynomial terms in the expansion. For models that require a high polynomial degree or are high-dimensional in terms of their input parameters, this number often exceeds the affordable computational cost. To overcome this issue, we apply a sparse Bayesian learning approach to the rational approximation. Through a specific prior distribution structure, sparsity is induced in the coefficients of the surrogate model. The denominator polynomial coefficients as well as the hyperparameters of the problem are determined through a type-II-maximum likelihood approach. We apply a quasi-Newton gradient-descent algorithm in order to find the optimal denominator coefficients and derive the required gradients through application of mathbbCR-calculus.