A lava attack on the recovery of sums of dense and sparse signals
Author(s)Liao, Yuan; Chernozhukov, Victor V; Hansen, Christian B.
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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.
DepartmentMassachusetts Institute of Technology. Department of Economics
The Annals of Statistics
Institute of Mathematical Statistics
Chernozhukov, Victor et al. “A Lava Attack on the Recovery of Sums of Dense and Sparse Signals.” The Annals of Statistics 45, 1 (February 2017): 39–76