On the uniqueness of weights in single-layer perceptrons.

In this paper the geometric formulation of the single layer perceptron weight optimization problem previously described by Coetzee et al. (1993, 1996) is combined with results from other researchers on nonconvex set projections to describe sufficient conditions for uniqueness of weight solutions. It is shown that the perceptron data surface is pseudoconvex and has infinite folding, allowing for the specification of a region of desired vectors having unique projections purely in terms of the local curvature of the data surface.

On a natural homotopy between linear and nonlinear single-layer networks.

In this paper we formulate a homotopy approach for solving for the weights of a network by smoothly transforming a linear single layer network into a nonlinear perceptron network. While other researchers have reported potentially useful numerical results based on heuristics related to this approach, the work presented here provides the first rigorous exposition of the deformation process. Results include a complete description of how the weights relate to the data space, a proof of the global convergence and validity of the method, and a rigorous formulation of the generalized orthogonality theorem to provide a geometric perspective of the solution process.