Storage capacity of perceptron with variable selection

Abstract

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

21 pages, 3 figures