A Tight Version of the Gaussian min-max theorem in the Presence of Convexity

Gaussian comparison theorems are useful tools in probability theory; they are essential ingredients in the classical proofs of many results in empirical processes and extreme value theory. More recently, they have been used extensively in the analysis of underdetermined linear inverse problems. A prominent role in the study of those problems is played by Gordon's Gaussian min-max theorem. It has been observed that the use of the Gaussian min-max theorem produces results that are often tight. Motivated by recent work due to M. Stojnic, we argue explicitly that the theorem is tight under additional convexity assumptions. To illustrate the usefulness of the result we provide an application example from the field of noisy linear inverse problems

Bilinearly indexed random processes -- \emph{stationarization} of fully lifted interpolation

Our companion paper \cite{Stojnicnflgscompyx23} introduced a very powerful \emph{fully lifted} (fl) statistical interpolating/comparison mechanism for bilinearly indexed random processes. Here, we present a particular realization of such fl mechanism that relies on a stationarization along the interpolating path concept. A collection of very fundamental relations among the interpolating parameters is uncovered, contextualized, and presented. As a nice bonus, in particular special cases, we show that the introduced machinery allows various simplifications to forms readily usable in practice. Given how many well known random structures and optimization problems critically rely on the results of the type considered here, the range of applications is pretty much unlimited. We briefly point to some of these opportunities as well