On the conditional distributions of low-dimensional projections from high-dimensional data
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高维数据低维投影的条件分布
- 研究问题
- 高维随机向量的低维投影在给定其他投影时，条件均值是否近似线性、条件方差是否近似恒定，这些近似覆盖哪些投影方向？
- 方法与思路
- 作者研究标准化高维随机向量的投影条件分布，在正则条件下分析均值与方差近似，并讨论方向上的一致性和覆盖范围。
- 材料陈述的贡献
- 摘要称补充了条件方差近似恒定的结果；当维度较大时，结论对响应方向一致、对多数解释方向成立，并讨论多种建模用途。
- 阅读关注点
- 应核对正则条件、近似误差的度量、对多数方向成立的准确含义，以及维度增长如何影响结论在具体降维模型中的使用。
- 阅读边界
- OpenAlex 索引摘要未注明稿件版本；摘要未给出误差界和维度要求，条件均值与方差的近似也不足以证明完整条件分布为高斯。
原文摘要
We study the conditional distribution of low-dimensional projections from high-dimensional data, where the conditioning is on other low-dimensional projections. To fix ideas, consider a random $d$-vector $Z$ that has a Lebesgue density and that is standardized so that $\mathbb{E} Z=0$ and $\mathbb{E} ZZ'=I_{d}$. Moreover, consider two projections defined by unit-vectors $\alpha$ and $\beta$, namely a response $y=\alpha'Z$ and an explanatory variable $x=\beta'Z$. It has long been known that the conditional mean of $y$ given $x$ is approximately linear in $x$, under some regularity conditions; cf. Hall and Li [Ann. Statist. 21 (1993) 867–889]. However, a corresponding result for the conditional variance has not been available so far. We here show that the conditional variance of $y$ given $x$ is approximately constant in $x$ (again, under some regularity conditions). These results hold uniformly in $\alpha$ and for most $\beta$’s, provided only that the dimension of $Z$ is large. In that sense, we see that most linear submodels of a high-dimensional overall model are approximately correct. Our findings provide new insights in a variety of modeling scenarios. We discuss several examples, including sliced inverse regression, sliced average variance estimation, generalized linear models under potential link violation, and sparse linear modeling.