标题： 金融中的高维学习 显示英文标题

标题： High-Dimensional Learning in Finance

摘要： 近年来，机器学习的进展在使用大型过度参数化模型进行金融预测方面显示了有希望的结果。 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两个关键方面。 首先，我证明了在随机傅里叶特征实现中的样本内标准化从根本上改变了基础的高斯核近似，将平移不变核替换为依赖训练集的替代方案。 其次，我建立了信息论下界，确定了无论估计器多么复杂，可靠学习不可能发生的情况。 对多项式下界的详细定量校准表明，使用典型的参数选择，例如12,000个特征，12个月的观测数据，以及R平方2-3%，要摆脱该下界所需的样本量超过25-30年的数据——远超出任何实际使用的滚动窗口。 因此，观察到的样本外成功必须源于低复杂度的伪象，而不是来自预期的高维机制。 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 two 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 establish information-theoretic lower bounds that identify when reliable learning is impossible no matter how sophisticated the estimator. A detailed quantitative calibration of the polynomial lower bound shows that with typical parameter choices, e.g., 12,000 features, 12 monthly observations, and R-square 2-3%, the required sample size to escape the bound exceeds 25-30 years of data--well beyond any rolling-window actually used. Thus, observed out-of-sample success must originate from lower-complexity artefacts rather than from the intended high-dimensional mechanism.