# Spectral Deconfounding via Perturbed Sparse Linear Models

arXiv: https://arxiv.org/abs/1811.05352 · HTML: https://mlsanity.com/paper/1811.05352/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Nov 2018.

## Built on by (most cited first, 13 of 13)

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- [Correcting Confounding via Random Selection of Background Variables](https://mlsanity.com/paper/2202.02150/) (2022)
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- [Substitute adjustment via recovery of latent variables](https://mlsanity.com/paper/2403.00202/) (2024)
- [Pre-processing with Orthogonal Decompositions for High-dimensional Explanatory Variables](https://mlsanity.com/paper/2106.09071/) (2021)
- [Spectrally Deconfounded Gradient Boosting](https://mlsanity.com/paper/2607.09371/) (2026)
- [LUCID: Learning Under Confounding for Inference and Discovery in Time Series](https://mlsanity.com/paper/2609.31315/) (2026)
- [Linear Regression Beyond Covariance: Latent Slope Spectra from Higher-Order Cumulants](https://mlsanity.com/paper/2610.17772/) (2026)

## Abstract (arXiv)

Standard high-dimensional regression methods assume that the underlying coefficient vector is sparse. This might not be true in some cases, in particular in presence of hidden, confounding variables. Such hidden confounding can be represented as a high-dimensional linear model where the sparse coefficient vector is perturbed. For this model, we develop and investigate a class of methods that are based on running the Lasso on preprocessed data. The preprocessing step consists of applying certain spectral transformations that change the singular values of the design matrix. We show that, under some assumptions, one can achieve the optimal ℓ1-error rate for estimating the underlying sparse coefficient vector. Our theory also covers the Lava estimator (Chernozhukov et al. [2017]) for a special model class. The performance of the method is illustrated on simulated data and a genomic dataset.