无标记投影寻踪中的搜索维度：子空间限制的标度律

中文摘要 AI 辅助

投影寻踪旨在寻找一个方向，沿该方向数据看起来最不服从高斯分布。当观测空间包含一个较大的高斯补空间时，经验目标函数可能被一个不携带信号的极小化方向所最小化，其经验峰度低至真实值水平。样本分割揭示而非修复了这一失败。添加与潜在机制无关的坐标会降低搜索性能，同时保持贝叶斯可恢复性不变。将搜索限制在已知前向算子的列空间内，可以在负峰度分支上精确消除该失败。从数据中估计主子空间是另一种替代方案。在一个受控的双分量模型中，前导充分标度在前向算子传递判别信息的增益上有所不同：协方差尖峰估计为$\varsigma^{-4}$，四阶矩搜索为$\varsigma^{-8}$。在固定搜索维度下，测得的阈值比塌缩到$n/p^2$，指数为$0.156$，接近预测的$1/8$。这是一个由充分界驱动的经验支持的标度，而非证明的渐近精确律。当搜索维度变化时，测得的指数为$0.325$，显著大于$1/8$，且测试范围未能确定其函数形式。交叉位置也依赖于校准和模型配置。在下游超额误差准则下，该标度基本消失。

英文摘要

Projection pursuit searches for a direction along which the data look least Gaussian. When the observation space contains a large Gaussian complement, the empirical objective can be minimized by a direction that carries no signal, with empirical kurtosis as low as at the truth. Sample splitting exposes rather than repairs this failure. Appending coordinates independent of the latent regime degrades the search while leaving Bayes recoverability unchanged. Restricting the search to the column space of a known forward operator removes the failure exactly on the negative-kurtosis branch. Estimating a principal subspace from the data is the alternative. In a controlled two-component model, the leading sufficient scalings differ in the gain with which the operator transmits the discriminant: $ς^{-4}$ for covariance-spike estimation and $ς^{-8}$ for fourth-moment search. At fixed search dimension, the measured threshold ratio collapses onto $n/p^2$ with exponent $0.156$, close to the predicted $1/8$. This is an empirically supported scaling motivated by sufficient bounds, not a proved asymptotically tight law. When the search dimension is varied, the measured exponent is $0.325$, substantially larger than $1/8$, and the tested range does not identify its functional form. The crossing location also depends on calibration and model configuration. Under a downstream excess-error criterion, the scaling largely disappears.

发表机构

• University of Cambridge（剑桥大学）

机构由 AI 辅助整理，请以论文原文为准。