Storage capacity of perceptron with variable selection

A central challenge in machine learning is to distinguish genuine structure from chance correlations in high-dimensional data. In this work, we address this issue for the perceptron, a foundational model of neural computation. Specifically, we investigate the relationship between the pattern load α and the variable selection ratio ρ for which a simple perceptron can perfectly classify P = α N random patterns by optimally selecting M = ρ N variables out of N variables. While the Cover--Gardner theory establishes that a random subset of ρ N dimensions can separate α N random patterns if and only if α < 2ρ, we demonstrate that optimal variable selection can surpass this bound by developing a method, based on the replica method from statistical mechanics, for enumerating the combinations of variables that enable perfect pattern classification. This not only provides a quantitative criterion for distinguishing true structure in the data from spurious regularities, but also yields the storage capacity of associative memory models with sparse asymmetric couplings.