  The weight space of the Ising perceptron in which a set of random patterns is
stored is examined using the generating function of the partition function
$\phi(n)=(1/N)\log [Z^n]$ as the dimension of the weight vector $N$ tends to
infinity, where $Z$ is the partition function and $[ ... ]$ represents the
configurational average. We utilize $\phi(n)$ for two purposes, depending on
the value of the ratio $\alpha=M/N$, where $M$ is the number of random
patterns. For $\alpha < \alpha_{\rm s}=0.833 ...$, we employ $\phi(n)$, in
conjunction with Parisi's one-step replica symmetry breaking scheme in the
limit of $n \to 0$, to evaluate the complexity that characterizes the number of
disjoint clusters of weights that are compatible with a given set of random
patterns, which indicates that, in typical cases, the weight space is equally
dominated by a single large cluster of exponentially many weights and
exponentially many small clusters of a single weight. For $\alpha > \alpha_{\rm
s}$, on the other hand, $\phi(n)$ is used to assess the rate function of a
small probability that a given set of random patterns is atypically separable
by the Ising perceptrons. We show that the analyticity of the rate function
changes at $\alpha = \alpha_{\rm GD}=1.245 ... $, which implies that the
dominant configuration of the atypically separable patterns exhibits a phase
transition at this critical ratio. Extensive numerical experiments are
conducted to support the theoretical predictions.
