Abstract
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
ϕ(n) = (1/N)log[Zn] as the dimension of
the weight vector N
tends to infinity, where Z
is the partition function and 
represents the configurational average. We utilize
ϕ(n)
for two purposes, depending on the value of the ratio
α = M/N, where
M is the number of
random patterns. For α<αs = 0.833..., we employ ϕ(n), in conjunction with Parisi's one-step replica symmetry breaking scheme in the limit of 
, 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
α>αs, on the
other hand, ϕ(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
α = αGD = 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.