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The weight space of the Ising perceptron in which a set of random patterns isstored 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 toinfinity, where $Z$ is the partition function and $[ ... ]$ represents theconfigurational average. We utilize $\phi(n)$ for two purposes, depending onthe value of the ratio $\alpha=M/N$, where $M$ is the number of randompatterns. For $\alpha < \alpha_{\rm s}=0.833 ...$, we employ $\phi(n)$, inconjunction with Parisi's one-step replica symmetry breaking scheme in thelimit of $n \to 0$, to evaluate the complexity that characterizes the number ofdisjoint clusters of weights that are compatible with a given set of randompatterns, which indicates that, in typical cases, the weight space is equallydominated by a single large cluster of exponentially many weights andexponentially many small clusters of a single weight. For $\alpha > \alpha_{\rms}$, on the other hand, $\phi(n)$ is used to assess the rate function of asmall probability that a given set of random patterns is atypically separableby the Ising perceptrons. We show that the analyticity of the rate functionchanges at $\alpha = \alpha_{\rm GD}=1.245 ... $, which implies that thedominant configuration of the atypically separable patterns exhibits a phasetransition at this critical ratio. Extensive numerical experiments areconducted to support the theoretical predictions.

Weight space structure and analysis using a finite replica number in the Ising perceptron