On the conditional distributions of low-dimensional projections from high-dimensional data

TL;DR

This paper demonstrates that in high-dimensional data, the conditional variance of a low-dimensional projection remains approximately constant given another projection, under certain regularity conditions, extending understanding of linear submodels.

Contribution

It provides the first theoretical result showing the approximate constancy of conditional variance in high-dimensional projections, broadening the scope of linear model assumptions.

Findings

Conditional mean is approximately linear in high dimensions.

Conditional variance is approximately constant in high dimensions.

Results hold uniformly for most projection directions.

Abstract

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.