The previous analysis requires a solution to two quadratic equations. This is of some concern should we wish to consider neural representations. Conveniently, however, equation ( 2 ) may be expressed as:
where the weights (
) explicitly reference the transparent velocities in each layer (eg
). A linear regression network that depicts this model is shown in
figure
2
a. In the network shown, each unit is linear. It is a recurrent
feedforward one where the desired response (output) is set to zero. To
train the network, we have used a linear version of the backpropagation
algorithm. We define the energy to this network's output (
) by:
where
refers to the output of the
i
th hidden unit. To train the network shown in figure 2a one can
minimise:
where
is a Lagrange multiplier. The convergence of the weights to minimise
equation (
11
) is equivalent to a least-squares regression. The network must be
trained using equations (6)-(8): either in serial or parallel. Parallel
training would require a modified learning rule. One could use the
recurrent backpropagation algorithm[
7
] to a network with three output nodes that represent equations (6)-(8).
In view of the analogy to multiple regression to these networks, we will
concentrate on solutions based upon the SVD given in appendix A.
The above representation has notable advantages for higher order transparencies. Here, Shizawa and Mase[ 10 ] show us how we may cascade derivative operators n times to detect an n -fold transparency. For a three-fold transparency (in 1-d) this leads to the constraint:
which is depicted as a neural network in figure 2 b. One should note that these neural networks conveniently represent one motion per layer. This has a considerable advantage for transparent models because we can avoid the need to solve quadratic, cubic and higher-order equations.
Adrian F Clark
