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

Author

Listed:
  • Victor Chernozhukov

    (Institute for Fiscal Studies and MIT)

  • Christian Hansen

    (Institute for Fiscal Studies and Chicago GSB)

  • Yuan Liao

    (Institute for Fiscal Studies)

Abstract

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 non-zero parameters that are large in magnitude, or a dense signal model, a model with no large parameters and very many small non-zero 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 feasible, data-dependent penalty parameters and illustrates lava’s improved performance relative to these benchmarks.

Suggested Citation

  • Victor Chernozhukov & Christian Hansen & Yuan Liao, 2015. "A lava attack on the recovery of sums of dense and sparse signals," CeMMAP working papers CWP05/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:05/15
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    References listed on IDEAS

    as
    1. Victor Chernozhukov & Christian Hansen & Yuan Liao, 2015. "A lava attack on the recovery of sums of dense and sparse signals," CeMMAP working papers 56/15, Institute for Fiscal Studies.
    2. Jianqing Fan & Yuan Liao & Martina Mincheva, 2013. "Large covariance estimation by thresholding principal orthogonal complements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 603-680, September.
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    6. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
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    1. Victor Chernozhukov & Christian Hansen & Yuan Liao, 2015. "A lava attack on the recovery of sums of dense and sparse signals," CeMMAP working papers 56/15, Institute for Fiscal Studies.

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    Keywords

    High-dimensional models; penalization; shrinkage; non-sparse signal recovery;
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