  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.
