# Inverse regression for ridge recovery: A data-driven approach for parameter reduction in computer experiments

## TL;DR

This paper interprets inverse regression methods SIR and SAVE as estimating ridge functions for parameter reduction in computer experiments, analyzing their convergence and limitations in identifying true important directions.

## Contribution

It provides a new interpretation of SIR and SAVE as ridge function estimators and analyzes their convergence and potential pitfalls in computer experiment contexts.

## Key findings

- SIR and SAVE estimate matrices whose column spaces contain ridge directions.
- Convergence of these column spaces improves with more data queries.
- SIR and SAVE may misidentify important directions when functions are not true ridge functions.

## 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.