2018/03/20 by Yuan Li, Benjamin Mark, Li, Yuan +9 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #Advanced Causal Inference Techniques #FOS: Computer and information sciences #Gene expression and cancer classification #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1803.07658
openalex publication_date 2018/03/20 · openalex created_date 2019/10/18 · openalex updated_date 2026/07/28
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 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.