Correlated variables in regression: Clustering and sparse estimation

Peter Bühlmann, Philipp Rütimann, Sara van de Geer, Cun Hui Zhang

Research output: Contribution to journalArticlepeer-review

107 Scopus citations


We consider estimation in a high-dimensional linear model with strongly correlated variables. We propose to cluster the variables first and do subsequent sparse estimation such as the Lasso for cluster-representatives or the group Lasso based on the structure from the clusters. Regarding the first step, we present a novel and bottom-up agglomerative clustering algorithm based on canonical correlations, and we show that it finds an optimal solution and is statistically consistent. We also present some theoretical arguments that canonical correlation based clustering leads to a better-posed compatibility constant for the design matrix which ensures identifiability and an oracle inequality for the group Lasso. Furthermore, we discuss circumstances where cluster-representatives and using the Lasso as subsequent estimator leads to improved results for prediction and detection of variables. We complement the theoretical analysis with various empirical results.

Original languageEnglish (US)
Pages (from-to)1835-1858
Number of pages24
JournalJournal of Statistical Planning and Inference
Issue number11
StatePublished - Nov 2013

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


  • Canonical correlation
  • Group Lasso
  • Hierarchical clustering
  • High-dimensional inference
  • Lasso
  • Oracle inequality
  • Variable screening
  • Variable selection


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