9 problems

Let $\mathbf{\mathcal S}(G,\kappa,\alpha)$ be a statistical symmetric binary perceptron, let $\alpha_{ult_s}(\kappa)$ be its $s$-level ultrametric OGP critical density, and let…

Let $\mathbf{\mathcal S}(G,\kappa,\alpha)$ be a statistical symmetric binary perceptron, let $\alpha_{ult_s}(\kappa)$ be its $s$-level ultrametric OGP critical density, let…

Let $\alpha_a$ denote the algorithmic constraint-density threshold for the statistical symmetric binary perceptron, let $\alpha_{ult_s}$ denote the critical density associated with…

Let $G\nin\mathbb{R}^{m\times n}$ have independent standard-normal entries. For constraint density $\alpha=\lim_{n\to\infty}m/n$ and $\kappa\in\mathbb{R}_+$, consider the statistic…

Let $A\in\mathbb{R}^{\alpha N\times N}$ have independent standard Gaussian entries, and consider the set of $\sigma\in\{-1,+1\}^N$ satisfying $A\sigma>0$ entrywise. The largest sol…

Consider clusters of solutions of a given size in the symmetric binary perceptron, and compare the clusters described by the planting procedure with those described by the full sol…

Let $\alpha$ denote the constraint density in the symmetric binary perceptron. Let $\Sigma(s)$ be the complexity as a function of the entropy $s$, with one curve obtained from the…

Let $\alpha$ denote the constraint density, let $\Sigma(s)$ be the complexity as a function of the entropy $s$ for clusters in the symmetric binary perceptron, and let $\kappa_0$ b…

Let $S(\bm{G},\kappa)$ be the solution set of the symmetric binary perceptron, and let $\bm{x}_0$ be a rare solution. Consider the local entropy of solutions around $\bm{x}_0$ as a…