A lava attack on the recovery of sums of dense and sparse signals

Victor Chernozhukov, Christian Hansen, Yuan Liao

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


Common high-dimensional methods for prediction rely on having either a sparse signal model, a model in which most parameters are zero and there are a small number of nonzero parameters that are large in magnitude, or a dense signal model, a model with no large parameters and very many small nonzero parameters. We consider a generalization of these two basic models, termed here a "sparse + dense" model, in which the signal is given by the sum of a sparse signal and a dense signal. Such a structure poses problems for traditional sparse estimators, such as the lasso, and for traditional dense estimation methods, such as ridge estimation.We propose a new penalization-based method, called lava, which is computationally efficient.With suitable choices of penalty parameters, the proposed method strictly dominates both lasso and ridge. We derive analytic expressions for the finite-sample risk function of the lava estimator in the Gaussian sequence model. We also provide a deviation bound for the prediction risk in the Gaussian regression model with fixed design. In both cases, we provide Stein's unbiased estimator for lava's prediction risk. A simulation example compares the performance of lava to lasso, ridge and elastic net in a regression example using data-dependent penalty parameters and illustrates lava's improved performance relative to these benchmarks.

Original languageEnglish (US)
Pages (from-to)39-76
Number of pages38
JournalAnnals of Statistics
Issue number1
StatePublished - Feb 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • High-dimensional models
  • Nonsparse signal recovery
  • Penalization
  • Shrinkage


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