# The Spectral Structure of Latent Treatment Effects

arXiv: https://arxiv.org/abs/2607.10926 · HTML: https://mlsanity.com/paper/2607.10926/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 55 references that are in the map.
- First posted on arXiv 12 Jul 2026.

## Builds on (most cited first, 5 of 5)

- [An Introduction to Proximal Causal Learning](https://mlsanity.com/paper/2009.10982/) (2020)
- [Counterexamples to "The Blessings of Multiple Causes" by Wang and Blei](https://mlsanity.com/paper/2001.06555/) (2020)
- [Causal Discovery in Mixtures of Populations](https://mlsanity.com/paper/2311.07454/) (2023)
- [Comparing Two Proxy Methods for Causal Identification](https://mlsanity.com/paper/2512.00175/) (2025)
- [Estimating Aleatoric Uncertainty in the Causal Treatment Effect](https://mlsanity.com/paper/2602.08461/) (2026)

## Abstract (arXiv)

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.