Fixed step size recursive algorithms for the covering problem

The authors present a family of techniques called adaptive covering algorithms (ACAs) that solve the problem of how to best cover a target shape using a set of simply parametrized elements. The algorithms, inspired by adaptive filtering techniques, provide a computationally simple, robust, and efficient alternative to more traditional methods such as Bayesian approaches, convex hulls, and multilayer perceptrons. The authors develop a theoretical understanding of the ACAs by relating their behavior to the evolution of a deterministic ordinary differential equation (ODE). Stability and instability of the ODE can be interpreted in terms of local stability/instability of the algorithm. In terms of the covering problem, candidate coverings tend to improve as more data are gathered whenever the ODE is stable. Several examples are given which demonstrate the ideas presented, and which verify that the analysis accurately predicts the true behavior of the algorithms.<>