Powerful rank verification for multivariate Gaussian data with any covariance structure

Upon observing $n$-dimensional multivariate Gaussian data, when can we infer that the largest $K$ observations came from the largest $K$ means? When $K=1$ and the covariance is isotropic, \cite{Gutmann} argue that this inference is justified when the two-sided difference-of-means test comparing the largest and second largest observation rejects. Leveraging tools from selective inference, we provide a generalization of their procedure that applies for both any $K$ and any covariance structure. We show that our procedure draws the desired inference whenever the two-sided difference-of-means test comparing the pair of observations inside and outside the top $K$ with the smallest standardized difference rejects, and sometimes even when this test fails to reject. Using this insight, we argue that our procedure renders existing simultaneous inference approaches inadmissible when $n > 2$. When the observations are independent (with possibly unequal variances) or equicorrelated, our procedure corresponds exactly to running the two-sided difference-of-means test comparing the pair of observations inside and outside the top $K$ with the smallest standardized difference.

A Statistical View of Column Subset Selection

We consider the problem of selecting a small subset of representative variables from a large dataset. In the computer science literature, this dimensionality reduction problem is typically formalized as Column Subset Selection (CSS). Meanwhile, the typical statistical formalization is to find an information-maximizing set of Principal Variables. This paper shows that these two approaches are equivalent, and moreover, both can be viewed as maximum likelihood estimation within a certain semi-parametric model. Using these connections, we show how to efficiently (1) perform CSS using only summary statistics from the original dataset; (2) perform CSS in the presence of missing and/or censored data; and (3) select the subset size for CSS in a hypothesis testing framework.