Why are we interested in piecewise rational functions?

Here are few reasons:
  1. When properly set over intervals such as #tex2html_wrap_inline2827#, piecewise rational functions can approximate the function #tex2html_wrap_inline2829# over the same interval.
  2. Evaluation of #tex2html_wrap_inline2831# functions may be much faster than evaluation of #tex2html_wrap_inline2833#.
  3. If training and convergence are not significantly worse for #tex2html_wrap_inline2835# nets then their overall training will be faster.
  4. For sequential implementations, the time to evaluate #tex2html_wrap_inline2837# (the activation function) and its first derivative, #tex2html_wrap_inline2839#, may not be very important in large networks; but parallel implementations can be more sensitive to the effort required to evaluate #tex2html_wrap_inline2841# and #tex2html_wrap_inline2843#. Thus, we expect a faster parallel implementation when #tex2html_wrap_inline2845# functions are used as activation functions.