标题： 金融领域的高维学习 显示英文标题

标题： High-Dimensional Learning in Finance

摘要： 机器学习领域的近期进展在使用大规模过参数化模型进行金融预测时展现了有前景的结果。 本文提供了理论基础和实证验证，以理解这些方法何时以及如何实现预测成功。 我研究了金融领域高维学习的三个关键方面。 首先，我证明了随机傅里叶特征实现中的样本内标准化从根本上改变了底层的高斯核近似，用依赖于训练集的替代方案取代了平移不变核。 其次，我推导了样本复杂度界限，表明在典型的金融弱信噪比下，可靠学习变得信息论上不可能的情况。 第三，VC维分析揭示了岭回归的复杂性由样本量而非名义特征维度决定。 全面的数值验证确认了这些理论预测，揭示了在现实参数范围内声称的理论性质的系统性失效。 这些结果表明，当样本量小且特征维度高时，观察到的预测成功必然由低复杂度的伪影驱动，而非真正的高维学习。 显示英文摘要

摘要： Recent advances in machine learning have shown promising results for financial prediction using large, over-parameterized models. This paper provides theoretical foundations and empirical validation for understanding when and how these methods achieve predictive success. I examine three key aspects of high-dimensional learning in finance. First, I prove that within-sample standardization in Random Fourier Features implementations fundamentally alters the underlying Gaussian kernel approximation, replacing shift-invariant kernels with training-set dependent alternatives. Second, I derive sample complexity bounds showing when reliable learning becomes information-theoretically impossible under weak signal-to-noise ratios typical in finance. Third, VC-dimension analysis reveals that ridgeless regression's effective complexity is bounded by sample size rather than nominal feature dimension. Comprehensive numerical validation confirms these theoretical predictions, revealing systematic breakdown of claimed theoretical properties across realistic parameter ranges. These results show that when sample size is small and features are high-dimensional, observed predictive success is necessarily driven by low-complexity artifacts, not genuine high-dimensional learning.