Examples

The following example shows a network in which the concepts of irreducibility and uniqueness do not hold. Again, the hidden and output layers of Figure-#fig8#699>(a) are composed of <#700#> clip<#700#> functions. Clearly n has a value greater than 1, hence #tex2html_wrap_inline3181#. Now, the net of Figure-#fig8#701>(a) has the following output:

#equation1881#

#equation1884#

#equation1887#

#equation1890#

#equation1893#

#equation1896#

Since n ;SPM_gt; 1 we get #tex2html_wrap_inline3185#, thus #tex2html_wrap_inline3187#.

#equation1899#

#equation1902#

The net of Figure-#fig8#718>(b) gives an output of:

#equation1905#

While the output of Figure-#fig8#720>(c) net is:

#equation1908#

#equation1911#

#figure723#
Figure 5: Three piecewise linear nets.

Since #tex2html_wrap_inline3189#, the three networks are I-O equivalent. Clearly, the nets are not structurely equivalent, which contradicts the concepts of irreducibility and uniqueness discussed in the last section. Hence, piecewise linear nets using clip activation functions, do not satisfy the properties of irreducibility and uniqueness.