The Spectral Structure of Latent Treatment Effects

2026-07-12

AI Summary

This study addresses the identification of heterogeneous treatment effects in the presence of unobserved discrete confounders. Under a population factorization assumption, the authors construct a compressed observable operator whose spectral structure—derived from the difference between two treatment-group quotient operators—directly reveals the latent treatment effects. The approach unifies higher-order moment estimation as bilinear functionals of operator powers, accommodating overcomplete proxy systems. By leveraging spectral analysis, operator similarity transformations, anchor normalization, and finite-dimensional perturbation theory, the method circumvents conventional inversion of high-order scalar moments. The proposed framework enables efficient estimation of treatment effects, feature matrices, and mixing proportions, and provides first-order high-probability perturbation bounds for these quantities.

Abstract

Identifying heterogeneous treatment effects under unobserved confounding is central in observational causal inference. In proxy models with a discrete latent confounder, prior Synthetic Potential Outcomes (SPO) [Mazaheri-Squires-Uhler '25] recover the mixture of treatment effects through recursively constructed scalar moments. We show that this sequence is one projection of a more fundamental object. Under the same population factorization assumptions, there is an exact compressed observable operator: after projecting onto the shared proxy signal subspace, the difference of two treatment-arm quotient operators is similar to the diagonal matrix of latent treatment effects. Its eigenvalues are the latent effects; its lifted left eigenvectors, after anchor normalization, recover the target-proxy feature matrix and then the latent mixture proportions. Every scalar SPO moment is a bilinear functional of a power of this operator. The resulting estimator handles overcomplete proxy systems, replaces high-order scalar inversion with finite-dimensional spectral analysis, and admits high-probability first-order perturbation bounds for treatment effects, feature rows, and simplex-projected mixture weights.

Problem

• heterogeneous treatment effects
• unobserved confounding
• latent confounder
• proxy models
• causal inference

Innovation

• spectral analysis
• latent treatment effects
• proxy variables
• observable operator
• heterogeneous treatment effects