论文标题
高维无序的对数孔孔吉布斯测量的强副本对称性
Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures
论文作者
论文摘要
我们考虑一类通用的对数孔库，可能是随机的（Gibbs）措施。我们证明了一个无限的阶参数家族的浓度，称为多verlap。因为它们完全参数了系统的猝灭吉布斯度量，所以这意味着渐近吉布斯测量值的简单表示，以及变量在很强的意义上的解耦。这些结果可能证明自己在多种情况下有用。尤其是在机器学习和高维推断中，对数洞穴措施出现在凸经验风险最小化，最大A-tosteriori推理或M估计中。我们认为，它们可能适用于在此类设置中为自由能，推理或泛化错误建立某种类型的“副本对称公式”。
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.
