A model of adaptive invariance
This thesis is about adaptive invariance, and a new model of it: the Group Representation Network. We begin by discussing the concept of adaptive invariance. We then present standard background material, mostly from the fields of group theory and neural networks. Following this we introduce the problem of invariant pattern recognition and describe a number of methods for solving various instances of it. Next, we define the Symmetry Network, a connectionist model of permutation invariance, and we develop some new theory of this model. We also extend the applicability of the Symmetry Network to arbitrary finite group actions. We then introduce the Group Representation Network (GRN) as an abstract model, with which in principle we can construct concomitants between arbitrary group representations. We show that the GRN can be regarded as a neural network model, and that it includes the Symmetry Network as a submodel. We apply group representation theory to the analysis of GRNs. This yields general characterizations of the allowable activation functions in a GRN and of their weight matrix structure. We examine various generalizations and restricted cases of the GRN model, and in particular look at the construction of GRNs over infinite groups. We then consider the issue of a GRN's discriminability, which relates to the problem of graph isomorphism. We look next at the computational abilities of the GRN, and postulate that it is capable of approximately computing any group concomitant. We show constructively that any polynomial concomitant can be computed by a GRN. We also prove that a variety of standard models for invariant pattern recognition can be viewed as special instances of the GRN model. Finally, we propose that the GRN model may be biologically plausible and give suggestions for further research.