The covering problem and μ-dependent adaptive algorithms

The paper presents a family of techniques, called adaptive covering algorithms, which solve a particular covering problem-how to best cover a target shape using a set of simply parameterized 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 multi-layer perceptrons. The paper develops a theoretical understanding of the adaptive covering algorithms by relating their behavior via weak convergence techniques to the evolution of a deterministic ordinary differential equation (ODE). In the process, the authors give new convergence results for a class of step size-dependent recursive algorithms. Stability and instability of the ODE can be interpreted in terms of the local stability/instability of the algorithm. In terms of the covering problem, candidate coverings tend to improve as more data is gathered whenever the ODE is stable. Several examples are given which demonstrate the ideas and which verify that the analysis accurately predicts the true behavior of the algorithms