Perturbative methods for non-parametric instrumental variable

2026-05-29

AI Summary

This study addresses the ill-posedness of integral equations arising from the curse of dimensionality in high-dimensional nonparametric instrumental variable (NPIV) estimation. For the first time, it introduces physical perturbation theory to this domain and proposes a kernel ridge regression–based higher-order perturbation correction method. By leveraging spectral analysis of integral operators and hybridization of eigenmodes, the approach systematically constructs a perturbation expansion that effectively mitigates high-dimensional ill-posedness while preserving computational tractability. Theoretical analysis and empirical experiments demonstrate that, with first-order correction, the method reduces prediction error by up to 99% under severe ill-posedness (β > 0.7), with performance gains intensifying as dimensionality increases—substantially outperforming existing approaches.

Abstract

We introduce a perturbative approach for nonparametric instrumental variable (NPIV) estimation. By drawing inspiration from perturbation theory in physics, we extend standard kernel ridge methods with systematic higher perturbation order corrections that significantly improve estimation accuracy. Spectrally, the perturbation introduces mixing between different eigenmodes of the expectation integral operator, which becomes especially useful when the integral equation is ill-defined. One source for such ill-definedness can be the curse of dimensionality. Our method performs across various dimensionality regimes, particularly when the dimensionality parameter $β$ which is defined through the number of samples $n$ and dimension $d$ as $n^β= d$, becomes large. Experimental results show that our first-order perturbative corrections can reduce prediction error by up to 99\% in high-dimensional ill-defined cases ($β> 0.7$) compared to standard ridge regression approaches. The performance improvement is maintained across a wide range of dimensions, with the advantage becoming more pronounced as dimensionality increases.

Problem

• nonparametric instrumental variable
• ill-posed inverse problem
• curse of dimensionality
• high-dimensional estimation

Innovation

• perturbative method
• nonparametric instrumental variable
• kernel ridge regression
• ill-posed integral equation
• curse of dimensionality