Definitions and Observations

(see [#12##1#]) A function #tex2html_wrap_inline3139# on the interval #tex2html_wrap_inline3141# is a piecewise linear (#tex2html_wrap_inline3143#) if there is a subdivision #tex2html_wrap_inline3145# such that #tex2html_wrap_inline3147# is linear on each interval #tex2html_wrap_inline3149#, for #tex2html_wrap_inline3151#. #tex2html_wrap_inline3153# In this paper, we only consider the class of #tex2html_wrap_inline3155# that meets the continuity criteria. This class will be referred to as the <#671#> clipping<#671#> type or the <#672#> clip<#672#> functions. The <#674#> clip<#674#> function (clip) is of one of the following two forms:

#equation1847#

#equation1859#

#tex2html_wrap_inline3159# The relationship between #tex2html_wrap_inline3161# and #tex2html_wrap_inline3163# is:

#equation1874#

#tex2html_wrap_inline3165# From this point on, any reference to <#693#> clip<#693#> is intended to refer to <#694#> #tex2html_wrap_inline3169#<#694#>. A piecewise linear network (#tex2html_wrap_inline3171#) is a feedforward multilayer net using #tex2html_wrap_inline3173# as activation functions. #tex2html_wrap_inline3175#