[Paper Review] A Lanczos-Stieltjes method for one-dimensional ridge function approximation and integration

Andrew Glaws, Paul G. Constantine|arXiv (Cornell University)|Aug 6, 2018

Probabilistic and Robust Engineering Design60 references4 citations

TL;DR

This paper introduces a Lanczos-Stieltjes method for efficient one-dimensional ridge function approximation and integration, leveraging repeated convolution to estimate the density of a linear combination of inputs and constructing orthogonal polynomials via Lanczos iteration. The method achieves exponential computational savings over full-dimensional approaches while maintaining high accuracy, especially under limited function evaluation budgets.

ABSTRACT

Many of the input-parameter-to-output-quantity-of-interest maps that arise in computational science admit a surprising low-dimensional structure, where the outputs vary primarily along a handful of directions in the high-dimensional input space. This type of structure is well modeled by a ridge function, which is a composition of a low-dimensional linear transformation with a nonlinear function. If the goal is to compute statistics of the output (e.g., as in uncertainty quantification or robust design) then one should exploit this low-dimensional structure, when present, to accelerate computations. We develop Gaussian quadrature and the associated polynomial approximation for one-dimensional ridge functions. The key elements of our method are (i) approximating the univariate density of the given linear combination of inputs by repeated convolutions and (ii) a Lanczos-Stieltjes method for constructing orthogonal polynomials and Gaussian quadrature.

Motivation & Objective

• To address the curse of dimensionality in high-dimensional integration by exploiting low-dimensional ridge structure in input-output maps.
• To develop a Gaussian quadrature-based method for one-dimensional ridge functions that reduces computational cost while preserving accuracy.
• To extend the method to functions approximately representable as ridge functions, enabling practical use in real-world scientific computing with expensive simulations.
• To provide a numerically stable and efficient algorithm for computing integrals and polynomial approximations of ridge functions using minimal function evaluations.

Proposed method

• Approximate the univariate density of the linear combination $ u = oldsymbol{a}^T oldsymbol{x} $ using repeated convolution of the input density, enabling accurate quadrature construction.
• Apply the Lanczos-Stieltjes algorithm to generate orthogonal polynomials and associated Gaussian quadrature rules for the induced measure on the ridge variable $ u $.
• Construct polynomial approximations of the ridge profile $ g(u) $ using the resulting quadrature nodes and weights, enabling accurate integration and function approximation.
• Extend the method to nearly ridge functions by using a two-level approximation: first approximating the input density $ q(u) $, then applying Lanczos iterations to build the polynomial surrogate.
• Use a shadow plot and $ L^2 $ error analysis to validate the ridge approximation quality and determine optimal polynomial degree.
• Leverage the nested structure of sparse grid-like rules and the hierarchical refinement of Lanczos iterations to enable convergence studies and adaptive refinement.

Experimental results

Research questions

• RQ1 Can a Lanczos-Stieltjes method be effectively adapted to construct Gaussian quadrature rules for one-dimensional ridge functions with minimal function evaluations?
• RQ2 How does the accuracy of one-dimensional ridge approximation compare to full-dimensional polynomial approximation under limited computational budgets?
• RQ3 To what extent can the method exploit low-dimensional structure in high-dimensional input-output maps to reduce integration cost?
• RQ4 What is the impact of approximating the induced density $ q(u) $ via convolution on the accuracy of the resulting quadrature and polynomial approximation?
• RQ5 How well can the method handle functions that are approximately, but not exactly, ridge functions?

Key findings

• The one-dimensional ridge approximation using the Lanczos-Stieltjes method achieves optimal performance with very few function evaluations, outperforming full-dimensional approximations under limited budgets.
• The $ L^2 $ error of the ridge approximation begins to level off at approximately $ 1.29 imes 10^{-2} $, indicating the limit of approximation accuracy due to the non-exactness of the ridge structure.
• The method achieves exponential computational savings compared to full-dimensional integration, particularly when the function exhibits strong ridge structure.
• The extension to nearly ridge functions is effective: the algorithm quickly approaches the inherent ridge approximation error with minimal function evaluations.
• The shadow plot and error analysis confirm that the method accurately captures the functional behavior along the ridge direction, even when the input is high-dimensional.
• The approach enables accurate integration of projection pursuit regression models by applying the method to each ridge component and summing the results, as shown in equation (57).