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Inverse regression for ridge recovery: A data-driven approach for parameter reduction in computer experiments
Numerical Analysis · Submitted 7 Feb 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Consequently, the computational scientist should beware when using SIR and SAVE for parameter reduction, since SIR and SAVE may mistakenly suggest that truly important directions are unimportant.
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The abstract, sentence by sentence
Parameter reduction can enable otherwise infeasible design and uncertainty studies with modern computational science models that contain several input parameters. In statistical regression, techniques for sufficient dimension reduction (SDR) use data to reduce the predictor dimension of a regression problem. A computational scientist hoping to use SDR for parameter reduction encounters a problem: a computer prediction is best represented by a deterministic function of the inputs, so data comprised of computer simulation queries fail to satisfy the SDR assumptions. To address this problem, we interpret SDR methods sliced inverse regression (SIR) and sliced average variance estimation (SAVE) as estimating the directions of a ridge function, which is a composition of a low-dimensional linear transformation with a nonlinear function. Within this interpretation, SIR and SAVE estimate matrices of integrals whose column spaces are contained in the ridge directions' span; we analyze and numerically verify convergence of these column spaces as the number of computer model queries increases. Moreover, we show example functions that are not ridge functions but whose inverse conditional moment matrices are low-rank. Consequently, the computational scientist should beware when using SIR and SAVE for parameter reduction, since SIR and SAVE may mistakenly suggest that truly important directions are unimportant.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Parameter reduction can enable otherwise infeasible design and uncertainty studies with modern computational science models that contain several input parameters. | background 1.00 | 0.05 | 0.03 | hedged possibility 0.60 |
| s1 | In statistical regression, techniques for sufficient dimension reduction (SDR) use data to reduce the predictor dimension of a regression problem. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s2 | A computational scientist hoping to use SDR for parameter reduction encounters a problem: a computer prediction is best represented by a deterministic function of the inputs, so data comprised of computer simulation queries fail to satisfy the SDR assumptions. | gap 0.56 | 0.07 | 0.04 | asserted 1.00 |
| s3 | To address this problem, we interpret SDR methods sliced inverse regression (SIR) and sliced average variance estimation (SAVE) as estimating the directions of a ridge function, which is a composition of a low-dimensional linear transformation with a nonlinear function. | contribution 0.55 | 0.78 | 0.09 | asserted 0.98 |
| s4 | Within this interpretation, SIR and SAVE estimate matrices of integrals whose column spaces are contained in the ridge directions' span; we analyze and numerically verify convergence of these column spaces as the number of computer model queries increases. | result 0.55 | 0.95 | 0.11 | asserted 1.00 |
| s5 | Moreover, we show example functions that are not ridge functions but whose inverse conditional moment matrices are low-rank. | result 0.93 | 0.87 | 0.05 | asserted 1.00 |
| s6 | Consequently, the computational scientist should beware when using SIR and SAVE for parameter reduction, since SIR and SAVE may mistakenly suggest that truly important directions are unimportant. | interpretation 0.74 | 0.41 | 0.74 | hedged possibility 0.99 |
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Original abstract
Parameter reduction can enable otherwise infeasible design and uncertainty studies with modern computational science models that contain several input parameters. In statistical regression, techniques for sufficient dimension reduction (SDR) use data to reduce the predictor dimension of a regression problem. A computational scientist hoping to use SDR for parameter reduction encounters a problem: a computer prediction is best represented by a deterministic function of the inputs, so data comprised of computer simulation queries fail to satisfy the SDR assumptions. To address this problem, we interpret SDR methods sliced inverse regression (SIR) and sliced average variance estimation (SAVE) as estimating the directions of a ridge function, which is a composition of a low-dimensional linear transformation with a nonlinear function. Within this interpretation, SIR and SAVE estimate matrices of integrals whose column spaces are contained in the ridge directions' span; we analyze and numerically verify convergence of these column spaces as the number of computer model queries increases. Moreover, we show example functions that are not ridge functions but whose inverse conditional moment matrices are low-rank. Consequently, the computational scientist should beware when using SIR and SAVE for parameter reduction, since SIR and SAVE may mistakenly suggest that truly important directions are unimportant.
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How the literature uses it
8 arXiv papers cite this in 15 sentences
Papers citing it · 11 sentences without a confident label
Cites it as background 4 papers
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Sufficient dimension reduction (SDR) provides a theoretical framework for dimension reduction in regression problems [ref] and has been used as a method for dimension reduction in deterministic functions [ref] .
Gauss-Christoffel quadrature for inverse regression: applications to computer experiments · Introduction and background · 0.85
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We point the interested reader to Sect. 2.2 of [ref] for further details on these differences and to the work of Glaws et al. [ref] , who provide a theoretical and computational framework for utilizing two well-known SDR techniques with computer models.
Supporting Multi-point Fan Design with Dimension Reduction · Statistical Sufficient Dimension Reduction · 0.96
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Dimensionality Reduction techniques [ref] include variable screening [ref] , kernel construction [ref] , Partial Least Squares (PLS), Principal Component Analysis (PCA), and sliced inverse regression (SIR).
Bayesian Calibration for Activity Based Models · Introduction · 0.89
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