Negative spherical perceptron

TL;DR

This paper extends the analysis of spherical perceptrons to the negative threshold case, providing new insights into its storage capacity and suggesting it may be more combinatorially complex than the positive case.

Contribution

It introduces a novel mechanism to analyze negative spherical perceptrons and offers the first detailed study of their storage capacity.

Findings

Negative spherical perceptrons may be more combinatorial in nature.

The analysis confirms the complexity of the negative case.

Results suggest the negative case is a harder challenge than the positive one.

Abstract

In this paper we consider the classical spherical perceptron problem. This problem and its variants have been studied in a great detail in a broad literature ranging from statistical physics and neural networks to computer science and pure geometry. Among the most well known results are those created using the machinery of statistical physics in \cite{Gar88}. They typically relate to various features ranging from the storage capacity to typical overlap of the optimal configurations and the number of incorrectly stored patterns. In \cite{SchTir02,SchTir03,TalBook} many of the predictions of the statistical mechanics were rigorously shown to be correct. In our own work \cite{StojnicGardGen13} we then presented an alternative way that can be used to study the spherical perceptrons as well. Among other things we reaffirmed many of the results obtained in \cite{SchTir02,SchTir03,TalBook} and thereby confirmed many of the predictions established by the statistical mechanics. Those mostly relate to spherical perceptrons with positive thresholds (which we will typically refer to as the positive spherical perceptrons). In this paper we go a step further and attack the negative counterpart, i.e. the perceptron with negative thresholds. We present a mechanism that can be used to analyze many features of such a model. As a concrete example, we specialize our results for a particular feature, namely the storage capacity. The results we obtain for the storage capacity seem to indicate that the negative case could be more combinatorial in nature and as such a somewhat harder challenge than the positive counterpart.