We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the ‘frozen 1-RSB’ structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1-RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.

中文翻译：

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对称伊辛感知器的冻结 1-RSB 结构

我们证明，在实值函数临界点的假设下，对称 Ising 感知器表现出 Krauth 和 Mézard 在物理文献中猜想的“冻结 1-RSB”结构；也就是说，模型的典型解位于熵密度消失的簇中。此外，我们以 Huang、Wong 和 Kabashima 猜想的非常有力的形式证明了这一点：模型的典型解以高概率被隔离，并且与所有其他解的汉明距离在维度上是线性的。冻结的 1-RSB 场景是 Baldassi、Ingrosso、Lucibello、Saglietti 和 Zecchina 最近对学习算法性能进行的有趣解释的一部分。我们通过将对称伊辛感知器模型与植入模型进行比较并证明两个模型之间的比较结果来证明这一结构结果。我们进行这种比较的主要技术工具是模型中解决方案数量对数集中度的归纳论证。