question about the definition of a polynomial
I am guessing that this is a bit past basic Algebra, but not quite advanced Algebra, so I will try here first. Please move this thread to the appropriate forum if I guessed incorrectly.
An artificial neural network can be reduced to an algebraic expressions with one variable/coefficient term for each network connection, one expression for the summation function on each node, and one expression for the transfer function on each node. In many ways, an ANN resembles a polynomial with many terms.
The exception is the log sigmoid transfer function,
yj = 1 / 1 + e-xj
where xj is the output of the summation function (linear sum of the inputs to node j from all nodes i). The exponent xj can be a real number and -xj can be negative. As far as I remember, both of these are contrary to the definition of a polynomial which are limited to positive integer exponents.
If an ANN expressed as an algebraic expression is not accurately described as a polynomial, what would the correct description be?
Re: question about the definition of a polynomial
I'm not clear what your question is. The way you have defined it, there is no reason to think that the transfer function must be any specific kind of function much less a polynomial. The example you give, , I would call simply an exponential function.