提出新方法，实现高维线性回归的无维度误差控制
Dimension-free bounds in high-dimensional linear regression via error-in-operator approach
- 通过误差算子法直接整合设计协方差，不显式估计Σ
- 得到非渐近的无维度误差上界，且余项可控
- 适用于高维数据，尤其适合参数调优后使用
研究随机设计下的高维线性回归问题。提出一种名为误差算子的新方法，不直接估计设计协方差Σ，而是将其融入经验风险最小化。我们推导了预测误差的展开式，并得到了首项和余项的非渐近无维度上界。结果表明，只要合理调整算法参数，辅助变量不会增加问题的有效维度。还讨论了方法的计算可行性，并通过数值实验验证了其性能。
原文摘要 · Abstract (English)
We consider a problem of high-dimensional linear regression with random design. We suggest a novel approach referred to as error-in-operator which does not estimate the design covariance $Σ$ directly but incorporates it into empirical risk minimization. We provide an expansion of the excess prediction risk and derive non-asymptotic dimension-free bounds on the leading term and the remainder. This helps us to show that auxiliary variables do not increase the effective dimension of the problem, provided that parameters of the procedure are tuned properly. We also discuss computational aspects of our method and illustrate its performance with numerical experiments.
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