Victor Chernozhukov, Christian Hansen, Yuan Liao
arXiv 11 Feb 2015 · Statistics — Methodology · publishedThe Annals of Statistics (2017) · 43 citations (OpenAlex)
arXiv:1502.03155 · PDF · DOI · OpenAlex · Extracted main text
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 an 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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Hsu, D., Kakade, S. M. and Zhang, T (2014) Random design analysis of ridge regression | 0.956 | 8 | 3 | 88% |
| 2 | Bickel, P., Ritov, Y. and Tsybakov, A (2009) Simultaneous analysis of lasso and dantzig selector | 0.874 | 8 | 2 | 100% |
| 3 | Belloni, A. and Chernozhukov, V (2013) Least squares after model selection in high-dimensional sparse models self | 0.874 | 5 | 2 | 100% |
| 4 | Stein, C. M (1981) Estimation of the mean of a multivariate normal distribution | 0.843 | 4 | 3 | 75% |
| 5 | Donoho, D. L. and Johnstone, I. M (1995) Adapting to unknown smoothness via wavelet shrinkage | 0.843 | 5 | 3 | 60% |
| 6 | Belloni, A., Chernozhukov, V. and Wang, L (2014) Pivotal estimation via square-root lasso in nonparametric regression self | 0.644 | 4 | 1 | 100% |
| 7 | Fan, J., Liao, Y. and Mincheva, M (2013) Large covariance estimation by thresholding principal orthogonal complements (with discussion) self | 0.644 | 2 | 2 | 100% |
| 8 | Candès, E. J., Li, X., Ma, Y. and Wright, J (2011) Robust principal component analysis? | 0.644 | 2 | 2 | 100% |
| 9 | Chandrasekaran, V., Sanghavi, S., Parrilo, P. A. and Willsky, A. S (2011) Rank-sparsity incoherence for matrix decomposition | 0.644 | 2 | 2 | 100% |
| 10 | Chen, Y. and Dalalyan, A (2012) Fused sparsity and robust estimation for linear models with unknown variance | 0.644 | 2 | 2 | 100% |
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