  We study the problem of determining the capacity of the binary perceptron for
two variants of the problem where the corresponding constraint is symmetric. We
call these variants the rectangle-binary-perceptron (RPB) and the
$u-$function-binary-perceptron (UBP). We show that, unlike for the usual
step-function-binary-perceptron, the critical capacity in these symmetric cases
is given by the annealed computation in a large region of parameter space (for
all rectangular constraints and for narrow enough $u-$function constraints,
$K<K^*$). We prove this fact (under two natural assumptions) using the first
and second moment methods. We further use the second moment method to
conjecture that solutions of the symmetric binary perceptrons are organized in
a so-called frozen-1RSB structure, without using the replica method. We then
use the replica method to estimate the capacity threshold for the UBP case when
the $u-$function is wide $K>K^*$. We conclude that full-step-replica-symmetry
breaking would have to be evaluated in order to obtain the exact capacity in
this case.
