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 $\alpha$ and the variable selection ratio $\rho$ for which a simple perceptron can perfectly classify $P = \alpha N$ random patterns by optimally selecting $M = \rho N$ variables out of $N$ variables. While the Cover--Gardner theory establishes that a random subset of $\rho N$ dimensions can separate $\alpha N$ random patterns if and only if $\alpha < 2\rho$, 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.