The inversion of a neural net tries to find an input pattern that
generates a specific output pattern with the existing connections. To
find this input, the deviation of each output from the desired output
is computed as error 
. This error value is used to approach
the target input in input space step by step. Direction and length of
this movement is computed by the inversion algorithm.
The most commonly used error value is the Least Mean Square
Error. 
is defined as
The goal of the algorithm therefore has to be to minimize 
.
Since the error signal 
can be computed as
and for the adaption value of the unit activation follows
In this implementation, a uniform pattern is applied to the input
units in the first step, whose activation level depends upon the
variable input pattern. This pattern is propagated through the
net and generates the initial output 
. The difference between
this output vector and the target output vector is propagated
backwards through the net as error signals 
. This is
analogous to the propagation of error signals in the backpropagation
training, with the difference that no weights are adjusted here. When
the error signals reach the input layer, they represent a gradient in
input space, which gives the direction for the gradient descent.
Thereby, the new input vector can be computed as
where 
is the step size in input space, which is set by the
variable eta.
This procedure is now repeated with the new input vector until the distance between the generated output vector and the desired output vector falls below the predefined limit of delta_max, when the algorithm is halted.
For a more detailed description of the algorithm and its implementation see [Mam92].