异质信号下的高维变量选择：精确渐近视角
High-dimensional variable selection with heterogeneous signals: A precise asymptotic perspective
AI总结：
本文研究异质微弱信号下高维稀疏回归的精确支撑恢复，证明BSS可达信息论最优阈值而边际筛选会失效，并提出计算高效且具模型一致性的两阶段ETS框架。
摘要：
我们研究在独立高斯设计下，当信号微弱、稀疏且可能具有异质性时，高维稀疏线性回归的精确支撑恢复问题。在样本量与信号稀疏度的适当标度下，我们将最小信号幅度固定在信息论最优速率，并考察最优子集选择（BSS）和边际筛选（MS）程序的渐近选择精度。我们表明，尽管处于理想设置，但有些令人惊讶的是，在存在异质信号时，边际筛选可能无法以趋于一的概率实现精确恢复；而只要最小信号强度高于信息论阈值，BSS便具有模型一致性。为缓解BSS的计算不可处理性，我们还提出了一个称为ETS（Estimate Then Screen，先估计后筛选）的高效两阶段算法框架，由估计步骤和梯度坐标筛选步骤组成；在关于样本量和稀疏度的相同标度假设下，我们证明ETS在与BSS相同的最小信号强度信息论最优要求下实现模型一致性。最后，我们给出了一项将ETS与LASSO和边际筛选进行比较的模拟研究。数值结果即使在样本量、维度和稀疏度取实际数值时也与我们的渐近理论相符。
英文摘要：
We study the problem of exact support recovery for high-dimensional sparse linear regression under independent Gaussian design when the signals are weak, rare, and possibly heterogeneous. Under a suitable scaling of the sample size and signal sparsity, we fix the minimum signal magnitude at the information-theoretic optimal rate and investigate the asymptotic selection accuracy of best subset selection (BSS) and marginal screening (MS) procedures. We show that despite the ideal setup, somewhat surprisingly, marginal screening can fail to achieve exact recovery with probability converging to one in the presence of heterogeneous signals, whereas BSS enjoys model consistency whenever the minimum signal strength is above the information-theoretic threshold. To mitigate the computational intractability of BSS, we also propose an efficient two-stage algorithmic framework called ETS (Estimate Then Screen) comprised of an estimation step and gradient coordinate screening step, and under the same scaling assumption on sample size and sparsity, we show that ETS achieves model consistency under the same information-theoretic optimal requirement on the minimum signal strength as BSS. Finally, we present a simulation study comparing ETS with LASSO and marginal screening. The numerical results agree with our asymptotic theory even for realistic values of the sample size, dimension and sparsity.
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