[Paper Review] Spectral Deconfounding via Perturbed Sparse Linear Models

Domagoj Ćevid, Peter Bühlmann+1 more|arXiv (Cornell University)|Nov 13, 2018

Statistical Methods and Inference33 references18 citations

TL;DR

This paper proposes a spectral deconfounding method that improves Lasso estimation in high-dimensional linear models with unobserved confounding by applying a perturbed sparse linear model framework. By transforming the design matrix via spectral transformations—particularly the novel Trim transform—it achieves optimal Lasso $β$-estimation error rates despite confounding, outperforming standard PCA adjustment and the Puffer transform in finite samples.

ABSTRACT

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 $\ell_1$-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.

Motivation & Objective

• Address the challenge of unobserved confounding in high-dimensional regression, where confounding induces dense perturbations to an otherwise sparse coefficient vector.
• Overcome limitations of standard Lasso in the presence of confounding, which leads to poor estimation and variable selection due to spurious correlations.
• Develop a general framework of spectral transformations that reweight the singular values of the design matrix to deconfound the regression problem.
• Theoretical justification is provided for achieving the optimal Lasso $β$-estimation error rate under confounding, even when the true coefficient vector is not exactly sparse.
• Demonstrate that the proposed method, especially the Trim transform, avoids the need to pre-specify the number of confounding components, unlike PCA-based adjustment.

Proposed method

• Apply a spectral transformation matrix $F$ to pre-process the design matrix $X$ and response vector $Y$, modifying the singular values of $X$ while preserving its column space structure.
• The transformation $F$ is constructed to shrink large singular values, with the Trim transform introduced as a novel method that sets singular values above a threshold to zero.
• The transformed data $(F Y, F X)$ are then fed into the Lasso for estimation, leveraging the fact that spectral pre-processing can mitigate the influence of confounding-induced correlations.
• Theoretical analysis relies on compatibility conditions and concentration inequalities, showing that the transformed design maintains favorable properties for high-dimensional estimation.
• The method is shown to be equivalent to the Lava estimator under a specific model class, extending its theoretical justification.
• Theoretical guarantees are derived using random matrix theory and bounds on the compatibility constant of the transformed design matrix.

Experimental results

Research questions

• RQ1 Can spectral transformations of the design matrix restore the optimal Lasso $β$-estimation error rate in the presence of confounding?
• RQ2 How does the performance of the proposed spectral deconfounding method compare to standard PCA adjustment and the Puffer transform?
• RQ3 Does the Trim transform eliminate the need to pre-specify the number of confounding components, as required in PCA-based methods?
• RQ4 Under what conditions does the compatibility constant of the transformed design matrix remain bounded away from zero, ensuring consistent estimation?
• RQ5 Can the proposed framework be theoretically linked to existing methods like the Lava estimator?

Key findings

• The Trim transform achieves the optimal Lasso $β$-estimation error rate under confounding, matching the minimax rate for sparse models.
• The method outperforms the Puffer transform in finite samples, particularly when the sample size is close to the number of predictors.
• Theoretical analysis shows that the compatibility constant of the transformed design matrix remains bounded away from zero with high probability, enabling consistent estimation.
• The Trim transform does not require estimating the number of confounding components, unlike PCA-based adjustment, which is a significant practical advantage.
• The method is theoretically equivalent to the Lava estimator under a specific model class, providing a new interpretation and justification for Lava in a spectral framework.
• Empirical results on simulated and genomic data confirm that the Trim transform improves variable selection and estimation accuracy under confounding, especially when the number of confounding components is unknown.