How to escape atypical regions in the symmetric binary perceptron: a journey through connected-solutions states

We study the binary symmetric perceptron model, and in particular its atypical solutions. While the solution-space of this problem is dominated by isolated configurations, it is also solvable for a certain range of constraint density $α$ and threshold $κ$. We provide in this paper a statistical measure probing sequences of solutions, where two consecutive elements shares a strong overlap. After simplifications, we test its predictions by comparing it to Monte-Carlo simulations. We obtain good agreement and show that connected states with a Markovian correlation profile can fully decorrelate from their initialization only for $κ>κ_{\rm no-mem.\, state}$ ($κ_{\rm no-mem.\, state}\sim \sqrt{0.91\log(N)}$ for $α=0.5$ and $N$ being the dimension of the problem). For $κ<κ_{\rm no-mem.\, state}$, we show that decorrelated sequences still exist but have a non-trivial correlations profile. To study this regime we introduce an $Ansatz$ for the correlations that we label as the nested Markov chain.