Abstract
This paper introduces a new concept of high-order parametric-shaped neurons in neural networks, and their application to geometric modeling of scattered points. These new neurons permit more effective modeling of data and point clusters with nonisotropic spatial distribution in multidimensional space compared to traditional networks and clustering methods. A high-order parametric-shape neuron is represented by a tensor that encapsulates correlation data from the activation signals associated with it. A traditional neuron is, in fact, a first-order case of the general high-order parametric- shape neuron. A second order neuron will learn not only the mean of the data cluster, but also its principal directions and its variance in these directions. The paper shows how high-order parametric-shaped neurons follow the maximum-correlation activation principle and permit simple Hebbian learning. A second-order neuron is fully formulated and demonstrated in a competitive network, where it performs successful self classification of complex spatial arrangements of data clusters. These arrangements include very close clusters and even partially overlapping clusters which are difficult to distinguish using traditional methods. The paper also discusses general mth-order neurons. An application of the proposed approach for geometrical modeling is discussed, where the network is used to model point clusters without making a priori assumptions regarding their topological configuration.