Convergence properties of symmetric learning algorithm for pattern classification

The geometric learning algorithm (GLA) was proposed as an application of th e affine projection algorithm for an adaptive filter to the perceptron. In the GLA, the connection weight vector w(n) is updated vertically toward the orthogonal complement of k patterns. The GLA demonstrates some typical beh avior when the learning rate lambda is 2, which means that w(n) and w(n + 1 ) are symmetric with respect to the complement. Therefore, in this paper, t he GLA with lambda = 2 is discriminated as the symmetric learning algorithm (SLA) and the convergence properties of the SLA are analyzed. The converge nce conditions regarding the order k of the SLA, the number P of patterns, and the dimension N of patterns are analyzed theoretically It is proved tha t k < N is the necessary condition for convergence when P > 2N. The relatio n between k and the learning speed is analyzed theoretically. It becomes cl ear that the maximum learning speed on average can be obtained when k = N/2 . These properties are supported by computer simulations. (C) 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(4): 18-25, 1999.