Fully lifted \emph{blirp} interpolation -- a large deviation view

TL;DR

This paper extends the fully lifted blirp interpolation framework using large deviation techniques, enabling analysis of both typical and atypical random structures, with applications to understanding solution clustering and computational gaps in complex optimization problems.

Contribution

It introduces a large deviation upgrade to the fully lifted blirp interpolation, broadening its applicability to atypical structures and connecting to local entropies in random optimization.

Findings

Wider applicability of the interpolation framework to atypical structures

Connections established between local entropies and solution clustering

Elegant final expressions for complex random models

Abstract

[104] introduced a powerful \emph{fully lifted} (fl) statistical interpolating mechanism. It established a nested connection between blirps (bilinearly indexed random processes) and their decoupled (linearly indexed) comparative counterparts. We here revisit the comparison from [104] and introduce its a \emph{large deviation} upgrade. The new machinery allows to substantially widen the [104]'s range of applicability. In addition to \emph{typical}, studying analytically much harder \emph{atypical} random structures features is now possible as well. To give a bit of a practical flavor, we show how the obtained results connect to the so-called \emph{local entropies} (LE) and their predicated role in understanding solutions clustering and associated \emph{computational gaps} in hard random optimization problems. As was the case in [104], even though the technical considerations often appear as fairly involved, the final interpolating forms admit elegant expressions thereby providing a relatively easy to use tool readily available for further studies. Moreover, as the considered models encompass all well known random structures discussed in [104], the obtained results automatically apply to them as well.