标题： 带有预测特征输入的核岭回归及其在因子非参数回归中的应用 显示英文标题

标题： Kernel Ridge Regression with Predicted Feature Inputs and Applications to Factor-Based Nonparametric Regression

摘要： 核方法，特别是核岭回归（KRR），是一种历经时间考验的强大非参数回归技术，以其丰富的容量、分析的简洁性以及计算的可行性而闻名。 对其预测性能的分析在过去二十多年间持续受到关注。 然而，在许多现代回归问题中，KRR所使用的特征输入无法直接观测，而是必须从其他测量值推断出来，KRR的理论基础仍然很大程度上未被探索。 本文介绍了一种针对使用预测特征输入的KRR的新分析方法。 我们的框架不仅对于处理预测特征输入至关重要，使我们能够在不对预测特征误差做任何假设的情况下得出风险界，而且通过允许任意模型误设，在平方损失下需要更弱的条件（特别是允许响应变量和函数类均无界），并且足够灵活以适应其他凸损失函数，从而加强了经典设定下的现有分析。 我们将一般理论应用于基于因子的非参数回归模型，并在使用主成分分析预测特征输入时建立了KRR的渐近最优性。 我们的理论发现通过模拟研究得到了进一步验证。 显示英文摘要

摘要： Kernel methods, particularly kernel ridge regression (KRR), are time-proven, powerful nonparametric regression techniques known for their rich capacity, analytical simplicity, and computational tractability. The analysis of their predictive performance has received continuous attention for more than two decades. However, in many modern regression problems where the feature inputs used in KRR cannot be directly observed and must instead be inferred from other measurements, the theoretical foundations of KRR remain largely unexplored. In this paper, we introduce a novel approach for analyzing KRR with predicted feature inputs. Our framework is not only essential for handling predicted feature inputs, enabling us to derive risk bounds without imposing any assumptions on the error of the predicted features, but also strengthens existing analyses in the classical setting by allowing arbitrary model misspecification, requiring weaker conditions under the squared loss, particularly allowing both an unbounded response and an unbounded function class, and being flexible enough to accommodate other convex loss functions. We apply our general theory to factor-based nonparametric regression models and establish the minimax optimality of KRR when the feature inputs are predicted using principal component analysis. Our theoretical findings are further corroborated by simulation studies.