Solution space structure of the Ising perceptron model

Consider the discrete cube $\{-1,1\}^N$ and a random collection of half spaces which includes each half space $H(x) := \{y \in \{-1,1\}^N: x \cdot y \geq \kappa \sqrt{N}\}$ for $x \in \{-1,1\}^N$ independently with probability $p$. The solution space is the intersection of these half spaces. We prove that the probability that the solution space is empty increases quickly from $\epsilon$ to $1 – \epsilon$ when $p$ increases only by a factor of $1 + o(1)$. I will also talk about a recent work with Will Perkins on the frozen structure of the solution space.

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