Self-Organised Factorial Encoding of a Toroidal Manifold
  It is shown analytically how a neural network can be used optimally to encode input data that is derived from a toroidal manifold. The case of a 2-layer network is considered, where the output is assumed to be a set of discrete neural firing events. The network objective function measures the average Euclidean error that occurs when the network attempts to reconstruct its input from its output. This optimisation problem is solved analytically for a toroidal input manifold, and two types of solution are obtained: a joint encoder in which the network acts as a soft vector quantiser, and a factorial encoder in which the network acts as a pair of soft vector quantisers (one for each of the circular subspaces of the torus). The factorial encoder is favoured for small network sizes when the number of observed firing events is large. Such self-organised factorial encoding may be used to restrict the size of network that is required to perform a given encoding task, and will decompose an input manifold into its constituent submanifolds. 
Stephen Luttrell