Ralf Der

Topology preservation in self-organizing feature maps: exact definition and measurement

The neighborhood preservation of self-organizing feature maps like the Kohonen map is an important property which is exploited in many applications. However, if a dimensional conflict arises this property is lost. Various qualitative and quantitative approaches are known for measuring the degree of topology preservation. They are based on using the locations of the synaptic weight vectors. These approaches, however, may fail in case of nonlinear data manifolds. To overcome this problem, in this paper we present an approach which uses what we call the induced receptive fields for determining the degree of topology preservation. We first introduce a precise definition of topology preservation and then propose a tool for measuring it, the topographic function. The topographic function vanishes if and only if the map is topology preserving. We demonstrate the power of this tool for various examples of data manifolds.

Building Nonlinear Data Models with Self-Organizing Maps

We study the extraction of nonlinear data models in high dimensional spaces with modified self-organizing maps. Our algorithm maps lower dimensional lattice into a high dimensional space without topology violations by tuning the neighborhood widths locally. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. The performance of the algorithm is demonstrated for one- and two-dimensional principal manifolds and for sparse data sets.

Classification in the Information Age

For some reinforcement learning algorithms the optimality of the generated strategies can be proven. In practice, however, restrictions in the number of training examples and computational resources corrupt optimality. The efficiency of the algorithms depends strikingly on the formulation of the task, including the choice of the learning parameters and the representation of the system states. We propose here to improve the learning efficiency by an adaptive classification of the system states which tends to group together states if they are similar and aquire the same action during learning. The approach is illustrated by two simple examples. Two further applications serve as a test of the proposed algorithm.

Higher Coordination With Less Control-A Result of Information Maximization in the Sensorimotor Loop

This work presents a novel learning method in the context of embodied artificial intelligence and self-organization, which has as few assumptions and restrictions as possible about the world and the underlying model. The learning rule is derived from the principle of maximizing the predictive information in the sensorimotor loop. It is evaluated on robot chains of varying length with individually controlled, noncommunicating segments. The comparison of the results shows that maximizing the predictive information per wheel leads to a higher coordinated behavior of the physically connected robots compared with a maximization per robot. Another focus of this article is the analysis of the effect of the robot chain length on the overall behavior of the robots. It will be shown that longer chains with less capable controllers outperform those of shorter length and more complex controllers. The reason is found and discussed in the information-geometric interpretation of the learning process.

Information driven self-organization of complex robotic behaviors.

Information theory is a powerful tool to express principles to drive autonomous systems because it is domain invariant and allows for an intuitive interpretation. This paper studies the use of the predictive information (PI), also called excess entropy or effective measure complexity, of the sensorimotor process as a driving force to generate behavior. We study nonlinear and nonstationary systems and introduce the time-local predicting information (TiPI) which allows us to derive exact results together with explicit update rules for the parameters of the controller in the dynamical systems framework. In this way the information principle, formulated at the level of behavior, is translated to the dynamics of the synapses. We underpin our results with a number of case studies with high-dimensional robotic systems. We show the spontaneous cooperativity in a complex physical system with decentralized control. Moreover, a jointly controlled humanoid robot develops a high behavioral variety depending on its physics and the environment it is dynamically embedded into. The behavior can be decomposed into a succession of low-dimensional modes that increasingly explore the behavior space. This is a promising way to avoid the curse of dimensionality which hinders learning systems to scale well.

Self-organized acquisition of situated behaviors

The paper aims at a systematic approach to the self-organization of behavior. It is rooted in the ideas of situated artificial intelligence and introduces situated behavior as the target for the self-organization procedure. Based on a quantitative measure of behavioral situatedness a learning dynamics is introduced which enables the controller to sustain the situatedness of the agent. The principle is demonstrated with Khepera robots in a number of different environmental conditions.

Information-driven self-organization: the dynamical system approach to autonomous robot behavior.

In recent years, information theory has come into the focus of researchers interested in the sensorimotor dynamics of both robots and living beings. One root for these approaches is the idea that living beings are information processing systems and that the optimization of these processes should be an evolutionary advantage. Apart from these more fundamental questions, there is much interest recently in the question how a robot can be equipped with an internal drive for innovation or curiosity that may serve as a drive for an open-ended, self-determined development of the robot. The success of these approaches depends essentially on the choice of a convenient measure for the information. This article studies in some detail the use of the predictive information (PI), also called excess entropy or effective measure complexity, of the sensorimotor process. The PI of a process quantifies the total information of past experience that can be used for predicting future events. However, the application of information theoretic measures in robotics mostly is restricted to the case of a finite, discrete state-action space. This article aims at applying the PI in the dynamical systems approach to robot control. We study linear systems as a first step and derive exact results for the PI together with explicit learning rules for the parameters of the controller. Interestingly, these learning rules are of Hebbian nature and local in the sense that the synaptic update is given by the product of activities available directly at the pertinent synaptic ports. The general findings are exemplified by a number of case studies. In particular, in a two-dimensional system, designed at mimicking embodied systems with latent oscillatory locomotion patterns, it is shown that maximizing the PI means to recognize and amplify the latent modes of the robotic system. This and many other examples show that the learning rules derived from the maximum PI principle are a versatile tool for the self-organization of behavior in complex robotic systems.

A new quantitative measure of topology preservation in Kohonen's feature maps

In this paper we give a new approach for quantifying topology preservation using explicitly the structure of the data manifold. It can be applied to linear and nonlinear data manifolds M. Further, this method allows one to quantify the range of folds. Our approach employs what we call the topographic function, which is defined based on the so called masked Voronoi polyhedra introduced by Martinetz (1993) for defining neighbourhood and topology preservation of feature maps.<<ETX>>

Guided self-organisation for autonomous robot development

The paper presents a method to guide the self-organised development of behaviours of autonomous robots. In earlier publications we demonstrated how to use the homeokinesis principle and dynamical systems theory to obtain self-organised playful but goal-free behaviour. Now we extend this framework by reinforcement signals. We validate the mechanisms with two experiment with a spherical robot. The first experiment aims at fast motion, where the robot reaches on average about twice the speed of a not reinforcement robot. In the second experiment spinning motion is rewarded and we demonstrate that the robot successfully develops pirouettes and curved motion which only rarely occur among the natural behaviours of the robot.

Rocking Stamper and Jumping Snakes from a Dynamical Systems Approach to Artificial Life

Dynamical systems offer intriguing possibilities as a substrate for the generation of behavior because of their rich behavioral complexity. However this complexity together with the largely covert relation between the parameters and the behavior of the agent is also the main hindrance in the goal oriented design of a behavior system. This paper presents a general approach to the self-regulation of dynamical systems so that the design problem is circumvented. We consider the controller (a neural net work) as the mediator for changes in the sensor values over time and define a dynamics for the parameters of the controller by maximizing the dynamical complexity of the sensorimotor loop under the condition that the consequences of the actions taken are still predictable. This very general principle is given a concrete mathematical formulation and is implemented in an extremely robust and versatile algorithm for the parameter dynamics of the controller. We consider two different applications, a mechanical device called the rocking stamper and the ODE simulations of a “snake” with five degrees of freedom. In these and many other examples studied we observed various behavior modes of high dynamical complexity