# Canonical correlation regression with noisy data

arXiv: https://arxiv.org/abs/2512.22697 · HTML: https://mlsanity.com/paper/2512.22697/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 37 references that are in the map.
- First posted on arXiv 27 Dec 2025.

## Builds on (most cited first, 8 of 8)

- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [Optimal Sup-norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression](https://mlsanity.com/paper/1508.03365/) (2015)
- [Causal Inference with Corrupted Data: Measurement Error, Missing Values, Discretization, and Differential Privacy](https://mlsanity.com/paper/2107.02780/) (2021)
- [Spectral Representation for Causal Estimation with Hidden Confounders](https://mlsanity.com/paper/2407.10448/) (2024)
- [Nonparametric Instrumental Variable Inference with Many Weak Instruments](https://mlsanity.com/paper/2505.07729/) (2025)
- [Demystifying Spectral Feature Learning for Instrumental Variable Regression](https://mlsanity.com/paper/2506.10899/) (2025)
- [Canonical Correlation Analysis: review](https://mlsanity.com/paper/2411.15625/) (2024)
- [Inference of high-dimensional weak instrumental variable regression models without ridge-regularization](https://mlsanity.com/paper/2504.20686/) (2025)

## Abstract (arXiv)

We study instrumental variable regression in data rich environments. The goal is to estimate a linear model from many noisy covariates and many noisy instruments. Our key assumption is that true covariates and true instruments are repetitive, though possibly different in nature; they each reflect a few underlying factors, however those underlying factors may be misaligned. We analyze a family of estimators based on two stage least squares with spectral regularization: canonical correlations between covariates and instruments are learned in the first stage, which are used as regressors in the second stage. As a theoretical contribution, we derive upper and lower bounds on estimation error, proving optimality of the method with noisy data. As a practical contribution, we provide guidance on which types of spectral regularization to use in different regimes.