[Paper Review] Graph-based regularization for regression problems with alignment and highly-correlated designs

Yuan Li, Benjamin Mark+4 more|arXiv (Cornell University)|Mar 20, 2018

Statistical Methods and Inference61 references4 citations

TL;DR

This paper proposes a graph-based total variation (GTV) regularization method for high-dimensional linear regression with highly correlated design matrices, leveraging a covariance graph to align feature correlations with regression coefficient similarities. The approach achieves optimal mean-squared error guarantees across diverse graph structures, including block and lattice graphs, and outperforms existing methods on synthetic and real biochemistry data.

ABSTRACT

Sparse models for high-dimensional linear regression and machine learning have received substantial attention over the past two decades. Model selection, or determining which features or covariates are the best explanatory variables, is critical to the interpretability of a learned model. Much of the current literature assumes that covariates are only mildly correlated. However, in many modern applications covariates are highly correlated and do not exhibit key properties (such as the restricted eigenvalue condition, restricted isometry property, or other related assumptions). This work considers a high-dimensional regression setting in which a graph governs both correlations among the covariates and the similarity among regression coefficients -- meaning there is \emph{alignment} between the covariates and regression coefficients. Using side information about the strength of correlations among features, we form a graph with edge weights corresponding to pairwise covariances. This graph is used to define a graph total variation regularizer that promotes similar weights for correlated features. This work shows how the proposed graph-based regularization yields mean-squared error guarantees for a broad range of covariance graph structures. These guarantees are optimal for many specific covariance graphs, including block and lattice graphs. Our proposed approach outperforms other methods for highly-correlated design in a variety of experiments on synthetic data and real biochemistry data.

Motivation & Objective

• Address the challenge of high-dimensional regression when design matrices exhibit high correlation among features, violating standard regularity assumptions like restricted eigenvalue conditions.
• Overcome identifiability issues in highly correlated designs by incorporating structural side information about feature correlations and coefficient similarities.
• Develop a regularization framework that leverages a graph derived from pairwise covariances to promote smoothness in estimated coefficients across correlated features.
• Establish theoretical mean-squared error guarantees for the proposed method under a novel alignment condition linking feature correlation structure and coefficient structure.
• Demonstrate empirical superiority of the method over existing approaches on synthetic data and real-world biochemistry datasets with highly correlated features.

Proposed method

• Construct a weighted graph from the empirical covariance matrix of the design matrix, where edge weights represent pairwise covariances between features.
• Define a graph total variation (GTV) regularizer that penalizes differences in regression coefficients across connected nodes in the graph, promoting similarity for correlated features.
• Formulate the optimization problem as a regularized least squares regression with the GTV penalty, enabling sparse and structured coefficient estimation.
• Introduce an alignment condition ensuring that the true coefficient vector β* respects the same graph structure as the feature correlations, resolving identifiability issues.
• Derive theoretical mean-squared error bounds for the GTV estimator under the alignment condition, showing optimality for block and lattice graph structures.
• Extend the framework to logistic regression in an appendix, demonstrating broader applicability beyond Gaussian noise models.

Experimental results

Research questions

• RQ1 Can graph-based regularization improve estimation accuracy in high-dimensional regression when features are highly correlated and standard regularity conditions fail?
• RQ2 How does incorporating a covariance-derived graph structure into the regularization process affect the theoretical error bounds of the estimator?
• RQ3 To what extent does the alignment between feature correlation structure and coefficient structure enhance model identifiability and performance?
• RQ4 What are the theoretical mean-squared error guarantees for the GTV estimator across different types of graph-structured covariance matrices?
• RQ5 How does the proposed method compare empirically to existing methods in terms of prediction accuracy on synthetic and real-world datasets with highly correlated features?

Key findings

• The proposed graph total variation (GTV) regularization achieves optimal mean-squared error rates for a broad class of covariance graph structures, including block and lattice graphs.
• Theoretical analysis shows that the GTV estimator attains minimax-optimal error rates under the proposed alignment condition, even when the design matrix lacks restricted eigenvalue or isometry properties.
• Empirical results on synthetic data demonstrate that GTV significantly outperforms Lasso, group Lasso, and fused Lasso in terms of estimation accuracy under high correlation.
• On real biochemistry data involving 242 chimeric P450 proteins, GTV achieves lower prediction error and better coefficient recovery compared to baseline methods, particularly when features are highly correlated.
• The method effectively leverages side information about feature correlations to improve model interpretability and robustness in high-dimensional settings.
• The extension to logistic regression in the appendix shows that the GTV framework is adaptable to non-Gaussian response models, broadening its practical applicability.