对密集和稀疏信号总和恢复的熔岩攻击
A lava attack on the recovery of sums of dense and sparse signals
摘要：
用于预测的常见的高维方法依赖于具有稀疏信号模型（其中大多数参数为零并且存在少量幅度大的非零参数的模型）或密集信号模型（其中没有大参数并且具有非常多的小的非零参数的模型）。我们考虑这两个基本模型的推广，这里称为“稀疏+密集”模型，其中信号由稀疏信号和密集信号的总和给出。这样的结构对传统的稀疏估计器（例如套索）和传统的密集估计方法（例如岭估计）提出了问题。我们提出了一种新的惩罚为基础的方法，称为熔岩，这是计算效率。通过适当选择惩罚参数，该方法严格控制套索和脊。在高斯序列模型中，我们推导出熔岩估计的有限样本风险函数的解析表达式。我们还提供了一个偏差界的预测风险的高斯回归模型与固定设计。在这两种情况下，我们都为熔岩的预测风险提供了Stein的无偏估计量。一个仿真示例比较了lava与lasso、ridge和弹性网在回归示例中使用可行的、依赖于数据的惩罚参数的性能，并说明了lava相对于这些基准的改进性能。
英文摘要：
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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