Geodesic Topographic Product: An Improvement to Measure Topology Preservation of Self-Organizing Neural Networks Francisco Flórez Revuelta, Juan Manuel García Chamizo, José García Rodríguez and Antonio Hernández Sáez Departamento de Tecnología Informática y Computación. Universidad de Alicante. Apdo. 99. 03080 Alicante, España {florez,juanma,jgarcia,ahernandez}@dtic.ua.es http://www.ua.es/i2rc Abstract. Self-organizing neural networks endeavour to preserve the topology of an input space by means of competitive learning. There are diverse measures that allow to quantify how good is this topology preservation. However, most of them are not applicable to measure non-linear input manifolds, since they don't consider the topology of the input space in their calculation. In this work, we have modified one of the most employed measures, the topographic prod- uct, incorporating the geodesic distance as distance measure among the refer- ence vectors of the neurons. Thus, it is possible to use it with non-lineal input spaces. This improvement allows to extend the studies realized with the origi- nal topographic product focused to the representation of objects by means of self-organizing neural networks. It would be also useful to determine the right dimensionality that a network must have to adapt correctly to an input mani- fold. 1 Introduction A self-organizing neural network A consists of a set of N neurons, each one of them has associated a reference vector belonging to the input space n i w ∈ R . These neurons are connected by edges to determine the topology of the network. By means of a competitive process, it is carried out an adaptation of the reference vectors as well as of the interconnection network among them; obtaining a mapping ψ that tries to preserve the topology of an input manifold n M ⊆ R , associating to each one of the input patterns M ξ ∈ its closest neuron i A ξ ∈ : , i j w w j A ξ ξ ξ − ≤ − ∀∈ (1) Researchers have usually considered this mapping as topology-preserving. How- ever, Martinetz and Schulten [1] limited this quality, by defining a Topology Repre-