On the uniqueness of weights in single-layer perceptrons

• Published in: IEEE transactions on neural networks Vol. 7; no. 2; pp. 318 - 325
• Main Authors: Coetzee, F.M., Stonick, V.L.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.1996; Institute of Electrical and Electronics Engineers
• Subjects: Applied sciences; Artificial intelligence; Computer science; control theory; systems; Connectionism. Neural networks; Exact sciences and technology; Hilbert space; Least squares methods; Manifolds; Neural networks; Sufficient conditions; Optimization; Geometrical projection; Neural network
• ISSN: 1045-9227; 1941-0093
• DOI: 10.1109/72.485635

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. No information is therefore required regarding the global curvature or size of the data surface. These results in principle allow for a posteriori evaluation of whether a weight solution is unique or globally optimal, and for a priori scaling of desired vector values to ensure uniqueness, through analysis of the input data. The practical applicability of these results from a numerical perspective is evaluated on some carefully chosen examples.