# Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures

arXiv: https://arxiv.org/abs/2009.12939 · HTML: https://mlsanity.com/paper/2009.12939/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Sep 2020.

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## Abstract (arXiv)

We consider a generic class of log-concave, possibly random, (Gibbs) measures. We prove the concentration of an infinite family of order parameters called multioverlaps. Because they completely parametrise the quenched Gibbs measure of the system, this implies a simple representation of the asymptotic Gibbs measures, as well as the decoupling of the variables in a strong sense. These results may prove themselves useful in several contexts. In particular in machine learning and high-dimensional inference, log-concave measures appear in convex empirical risk minimisation, maximum a-posteriori inference or M-estimation. We believe that they may be applicable in establishing some type of "replica symmetric formulas" for the free energy, inference or generalisation error in such settings.