#tex2html_wrap_inline3093#.
#equation1829#
otherwise
#equation1833#
Select a set of coefficients #tex2html_wrap_inline3095#. We need to show:
#equation1836#
By confining one's attention to x for which |x| is sufficiently
large that #tex2html_wrap_inline3101#, for all i, one can see that it is
sufficient to show that #tex2html_wrap_inline3105# satisfies IP. A sketch of the proof
follows:
<#1840#> Proof<#1840#>.\
<#651#>Note<#651#>:
We show by contradiction that #tex2html_wrap_inline3107# satisfies the IP. The proof is a
bit long and involved so we have decided not to include it
here. Furthermore, it is quite trivial to show that #tex2html_wrap_inline3109# satisfies
WIP. For more information on these concepts or the proof, contact the
author.
%
<#652#> Second<#652#>: Let the activation function be #tex2html_wrap_inline3111#.
In a personal communication, Sontag [#20##1#] said
that #tex2html_wrap_inline3113# satisfies the independence property.
Clearly, networks using #tex2html_wrap_inline3115#, and assuming the above conjecture,
networks using #tex2html_wrap_inline3117#, satisfy the property that
if #tex2html_wrap_inline3119# and #tex2html_wrap_inline3121# are irreducible in #tex2html_wrap_inline3123# and
the #tex2html_wrap_inline3125# then #tex2html_wrap_inline3127#.