Geometry and Expressive Power of Conditional Restricted Boltzmann Machines

Conditional restricted Boltzmann machines are undirected stochastic neural networks with a layer of input and output units connected bipartitely to a layer of hidden units. These networks define models of conditional probability distributions on the states of the output units given the states of the input units, parametrized by interaction weights and biases. We address the representational power of these models, proving results their ability to represent conditional Markov random fields and conditional distributions with restricted supports, the minimal size of universal approximators, the maximal model approximation errors, and on the dimension of the set of representable conditional distributions. We contribute new tools for investigating conditional probability models, which allow us to improve the results that can be derived from existing work on restricted Boltzmann machine probability models.Comment: 30 pages, 5 figures, 1 algorith

Evaluating Morphological Computation in Muscle and DC-motor Driven Models of Human Hopping

In the context of embodied artificial intelligence, morphological computation refers to processes which are conducted by the body (and environment) that otherwise would have to be performed by the brain. Exploiting environmental and morphological properties is an important feature of embodied systems. The main reason is that it allows to significantly reduce the controller complexity. An important aspect of morphological computation is that it cannot be assigned to an embodied system per se, but that it is, as we show, behavior- and state-dependent. In this work, we evaluate two different measures of morphological computation that can be applied in robotic systems and in computer simulations of biological movement. As an example, these measures were evaluated on muscle and DC-motor driven hopping models. We show that a state-dependent analysis of the hopping behaviors provides additional insights that cannot be gained from the averaged measures alone. This work includes algorithms and computer code for the measures.Comment: 10 pages, 4 figures, 1 table, 5 algorithm

A Theory of Cheap Control in Embodied Systems

We present a framework for designing cheap control architectures for embodied agents. Our derivation is guided by the classical problem of universal approximation, whereby we explore the possibility of exploiting the agent's embodiment for a new and more efficient universal approximation of behaviors generated by sensorimotor control. This embodied universal approximation is compared with the classical non-embodied universal approximation. To exemplify our approach, we present a detailed quantitative case study for policy models defined in terms of conditional restricted Boltzmann machines. In contrast to non-embodied universal approximation, which requires an exponential number of parameters, in the embodied setting we are able to generate all possible behaviors with a drastically smaller model, thus obtaining cheap universal approximation. We test and corroborate the theory experimentally with a six-legged walking machine. The experiments show that the sufficient controller complexity predicted by our theory is tight, which means that the theory has direct practical implications. Keywords: cheap design, embodiment, sensorimotor loop, universal approximation, conditional restricted Boltzmann machineComment: 27 pages, 10 figure

Morphological Computation: Synergy of Body and Brain

There are numerous examples that show how the exploitation of the body’s physical properties can lift the burden of the brain. Examples include grasping, swimming, locomotion, and motion detection. The term Morphological Computation was originally coined to describe processes in the body that would otherwise have to be conducted by the brain. In this paper, we argue for a synergistic perspective, and by that we mean that Morphological Computation is a process which requires a close interaction of body and brain. Based on a model of the sensorimotor loop, we study a new measure of synergistic information and show that it is more reliable in cases in which there is no synergistic information, compared to previous results. Furthermore, we discuss an algorithm that allows the calculation of the measure in non-trivial (non-binary) systems