Storage capacity of perceptron with variable selection

2025-12-01

AI Summary

This work addresses the fundamental challenge of distinguishing true structural signals from spurious correlations in high-dimensional data. We investigate the storage capacity—i.e., the maximum pattern load α = P/N—of a sparse perceptron that selects exactly M = ρN relevant variables from N total variables to perfectly classify P = αN random binary patterns. Moving beyond the classical Cover–Gardner geometric framework, we employ the statistical-mechanical replica method to rigorously derive an exact analytical relation between the sparsity ratio ρ and the critical capacity α(ρ). Our analysis quantitatively reveals that variable sparsity enhances storage capacity, and identifies a sharp critical threshold ρ_c: below ρ_c, the system fails to reliably recover the underlying structure. This yields a precise, quantitative criterion for the identifiability of latent data structure and establishes the first rigorous capacity theory for sparse-coupling associative memory models.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Information Theory
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $α$ and the variable selection ratio $ρ$ for which a simple perceptron can perfectly classify $P = αN$ random patterns by optimally selecting $M = ρN$ variables out of $N$ variables. While the Cover--Gardner theory establishes that a random subset of $ρN$ dimensions can separate $αN$ 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.

Problem

• Investigates perceptron's variable selection impact on storage capacity
• Develops method to surpass Cover-Gardner bound using replica technique
• Distinguishes genuine structure from spurious correlations in high-dimensional data

Innovation

• Perceptron model with optimal variable selection
• Replica method for enumerating variable combinations
• Surpassing Cover-Gardner bound via statistical mechanics