Conditional Means of Low-Dimensional Projections from High-Dimensional Data: Explicit Error Bounds

• Activity Number: 178
• Type: Contributed
• Date/Time: Monday, August 10, 2015 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Ivana Milovic* and Hannes Leeb
• Companies: University of Vienna and University of Vienna
• Keywords: conditional distributions ; high-dimensional linear model ; polynomial rate of convergence ; error bounds

Abstract

Take a random $d$-vector $Z$ that has a Lebesgue density and so that $E Z =0$ and $E Z Z' = I_d$. Consider two projections defined by unit-vectors $\alpha$ and $\beta$, namely a response $y = \alpha' Z$ and an explanatory variable $x = \beta' Z$. Under regularity conditions, Leeb has shown (2013, AoS) that for most $\beta$'s, $E[y|x]\approx$ linear in $x$, and that $Var[y|x]\approx$ const in $x$, provided that $d$ is large. These results imply that most simple submodels of a high-dimensional linear model are approximately correct. But Leeb's results are asymptotic, as $d\to \infty$ and no explicit bounds have been established. We provide explicit, finite-$d$ error bounds for the results regarding the conditional expectation. For a fixed $d$, let $E_d$ be the set of $\beta$'s such that $E[y|x]\approx \mbox{linear in } x$. We find bounds on the size of $E_d$ (as measured with respect to the uniform distribution $\upsilon$ on the unit $d$-sphere) and we show that its size increases very fast. Namely, $\upsilon(E_d)\to 1,$ as $d\to \infty$, at an arbitrary polynomial rate. We then apply our findings to the class of subgaussian random variables, to obtain even better results.