Looking at the basic model for a neural network, it is natural to consider: What is the simplest possible neural network? And is such a network useful for anything?
If we have a single layer (), only one node in the output layer () and no activation function ( is the identity function), then we have a very simple neural network. Obviously, having only one input node () will be the simplest, but we will initially consider any number of input nodes .
With this structure and input to the network given by we can compute the output by
(Here we have included a bias , as this was also included in the general model.)
So the output of the network is a linear combination of the input values (and the constant ). Furthermore, as seen from the post on the optimization problem, the error function is a least squares error function.
This means that the simple network described above is equivalent to linear regression (with a least squares error function, as is the most common).
This means that