Dominik Endres

A new metric for probability distributions

We introduce a metric for probability distributions, which is bounded, information-theoretically motivated, and has a natural Bayesian interpretation. The square root of the well-known /spl chi//sup 2/ distance is an asymptotic approximation to it. Moreover, it is a close relative of the capacitory discrimination and Jensen-Shannon divergence.

Bayesian bin distribution inference and mutual information

We present an exact Bayesian treatment of a simple, yet sufficiently general probability distribution model. We consider piecewise-constant distributions P(X) with uniform (second-order) prior over location of discontinuity points and assigned chances. The predictive distribution and the model complexity can be determined completely from the data in a computational time that is linear in the number of degrees of freedom and quadratic in the number of possible values of X. Furthermore, exact values of the expectations of entropies and their variances can be computed with polynomial effort. The expectation of the mutual information becomes thus available, too, and a strict upper bound on its variance. The resulting algorithm is particularly useful in experimental research areas where the number of available samples is severely limited (e.g., neurophysiology). Estimates on a simulated data set provide more accurate results than using a previously proposed method.

An application of formal concept analysis to semantic neural decoding

This paper proposes a novel application of Formal Concept Analysis (FCA) to neural decoding: the semantic relationships between the neural representations of large sets of stimuli are explored using concept lattices. In particular, the effects of neural code sparsity are modelled using the lattices. An exact Bayesian approach is employed to construct the formal context needed by FCA. This method is explained using an example of neurophysiological data from the high-level visual cortical area STSa. Prominent features of the resulting concept lattices are discussed, including indications for hierarchical face representation and a product-of-experts code in real neurons. The robustness of these features is illustrated by studying the effects of scaling the attributes.

Segmenting sign language into motor primitives with Bayesian binning

The endpoint trajectories of human movements fulfill characteristic power laws linking velocity and curvature. The parameters of these power laws typically vary between different segments of longer action sequences. These parameters might thus be exploited for the unsupervised segmentation of actions into movement primitives. For the example of sign language we investigate whether such segments can be identified by Bayesian binning (BB), using a Gaussian observation model whose mean has a polynomial time dependence. We show that this method yields good segmentation and correctly models ground truth kinematics composed of consecutive segments derived from wrist trajectories recorded from users of Israeli Sign Language (ISL). Importantly, polynomial orders between 3 and 5 yield an optimal trade-off between complexity and accuracy of the trajectory approximation, in accordance with the minimum acceleration and minimum jerk models. Comparing the orders of the polynomials best approximating natural kinematics against those needed to fit the power law ground truth data suggests that kinematic properties not compatible with power laws are also not adequately represented by low order polynomials and require higher order polynomials for a good approximation.

Learning movement primitives from optimal and dynamically feasible trajectories for humanoid walking

We present a new approach for humanoid gait generation based on movement primitives learned from optimal and dynamically feasible motion trajectories. As testing platform we consider the humanoid robot HRP-2, so far only in simulation. Training data is generated by solving a set of optimal control problems for a minimum-torque optimality criterion and five different step lengths. As the dynamic robot model with all its kinematic and dynamic constraints is considered in the optimal control problem formulation, the resulting motion trajectories are not only optimal but also dynamically feasible. For the learning process we consider the joint angle trajectories of all actuated joints, the ZMP trajectory and the pelvis trajectory, which are sufficient quantities to control the robot. From the training data we learn morphable movement primitives based on Gaussian processes and principal component analysis. We show that five morphable primitives are sufficient to generate steps with 24 different lengths, which are close enough to both dynamical feasibility and optimality to be useful for fast on-line movement generation.

Overtone-based pitch selection in hermit thrush song: Unexpected convergence with scale construction in human music

Significance The song of the hermit thrush, a common North American songbird, is renowned for its apparent musicality and has attracted the attention of musicians and ornithologists for more than a century. Here we show that hermit thrush songs, like much human music, use pitches that are mathematically related by simple integer ratios and follow the harmonic series. Our findings add to a small but growing body of research showing that a preference for small-integer ratio intervals is not unique to humans and are thus particularly relevant to the ongoing nature/nurture debate about whether musical predispositions such as the preference for consonant intervals are biologically or culturally driven. Many human musical scales, including the diatonic major scale prevalent in Western music, are built partially or entirely from intervals (ratios between adjacent frequencies) corresponding to small-integer proportions drawn from the harmonic series. Scientists have long debated the extent to which principles of scale generation in human music are biologically or culturally determined. Data from animal “song” may provide new insights into this discussion. Here, by examining pitch relationships using both a simple linear regression model and a Bayesian generative model, we show that most songs of the hermit thrush (Catharus guttatus) favor simple frequency ratios derived from the harmonic (or overtone) series. Furthermore, we show that this frequency selection results not from physical constraints governing peripheral production mechanisms but from active selection at a central level. These data provide the most rigorous empirical evidence to date of a bird song that makes use of the same mathematical principles that underlie Western and many non-Western musical scales, demonstrating surprising convergence between human and animal “song cultures.” Although there is no evidence that the songs of most bird species follow the overtone series, our findings add to a small but growing body of research showing that a preference for small-integer frequency ratios is not unique to humans. These findings thus have important implications for current debates about the origins of human musical systems and may call for a reevaluation of existing theories of musical consonance based on specific human vocal characteristics.

A novel approach for the generation of complex humanoid walking sequences based on a combination of optimal control and learning of movement primitives

We combine optimal control and movement primitive learning in a novel way for the fast generation of humanoid walking movements and demonstrate our approach at the example of the humanoid robot HRP-2 with 36 degrees of freedom. The present framework allows for an efficient computation of long walking sequences consisting of feasible steps of different kind: starting steps from a static posture, cyclic steps or steps with varying step lengths, and stopping motions back to a static posture. Together with appropriate sensors and high level decision strategies this approach provides an excellent basis for an adaptive walking generation on challenging terrain. Our framework comprises a movement primitive model learned from a small number of example steps that are dynamically feasible and minimize an integral mean of squared torques. These training steps are computed by solving three different kinds of optimal control problems that are restricted by the whole-body dynamics of the robot and the gait cycle. The movement primitive model decomposes the joint angles, pelvis orientation and ZMP trajectories in the example data into a small number of primitives, which effectively deals with the redundancy inherent in highly articulated motion. New steps can be composed by weighted combinations of these primitives. The mappings from step parameters to weights are learned with a Gaussian process approach, the contiguity of subsequent steps is promoted by conditioning the beginning of a new step on the end of the current one. Each step can be generated in less than a second, because the expensive optimal control computations, which take several hours per step, are shifted to the precomputational off-line phase. We validate our approach in the virtual robot simulation environment OpenHRP and study the effects of different kernels and different numbers of primitives. We show that the robot can execute long walking sequences with varying step lengths without falling, and hence that feasibility is transferred from optimized to generated motions. Furthermore, we demonstrate that the generated motions are close to torque optimality on the interior parts of the steps but have higher torques than their optimized counterparts on the steps boundaries. Having passed the validation in the robot simulator, we plan to tackle the transfer of this approach to the real platform HRP-2 as a next step.

Emulating human observers with bayesian binning: Segmentation of action streams

Natural body movements arise in the form of temporal sequences of individual actions. During visual action analysis, the human visual system must accomplish a temporal segmentation of the action stream into individual actions. Such temporal segmentation is also essential to build hierarchical models for action synthesis in computer animation. Ideally, such segmentations should be computed automatically in an unsupervised manner. We present an unsupervised segmentation algorithm that is based on Bayesian Binning (BB) and compare it to human segmentations derived from psychophysical data. BB has the advantage that the observation model can be easily exchanged. Moreover, being an exact Bayesian method, BB allows for the automatic determination of the number and positions of segmentation points. We applied this method to motion capture sequences from martial arts and compared the results to segmentations provided by humans from movies that showed characters that were animated with the motion capture data. Human segmentation was then assessed by an interactive adjustment paradigm, where participants had to indicate segmentation points by selection of the relevant frames. Results show a good agreement between automatically generated segmentations and human performance when the trajectory segments between the transition points were modeled by polynomials of at least third order. This result is consistent with theories about differential invariants of human movements.

Why Harmless Sensations Might Hurt in Individuals with Chronic Pain: About Heightened Prediction and Perception of Pain in the Mind

In individuals with chronic pain harmless bodily sensations can elicit anticipatory fear of pain resulting in maladaptive responses such as taking pain medication. Here, we aim to broaden the perspective taking into account recent evidence that suggests that interoceptive perception is largely a construction of beliefs, which are based on past experience and that are kept in check by the actual state of the body. Taking a Bayesian perspective, we propose that individuals with chronic pain display a heightened prediction of pain [prior probability p(pain)], which results in heightened pain perception [posterior probability p(pain|sensation)] due to an assumed link between pain and a harmless bodily sensation [p(sensation|pain)]. This pain perception emerges because their mind infers pain as the most likely cause for the sensation. When confronted with a mismatch between predicted pain and a (harmless bodily) sensation, individuals with chronic pain try to minimize the mismatch most likely by active inference of pain or alternatively by an attentional shift away from the sensation. The active inference results in activities that produce a stronger sensation that will match with the prediction, allowing subsequent perceptual inference of pain. Here, we depict heightened pain perception in individuals with chronic pain by reformulating and extending the assumptions of the interoceptive predictive coding model from a Bayesian perspective. The review concludes with a research agenda and clinical considerations.

Coupling Gaussian Process Dynamical Models with Product-of-Experts Kernels

We describe a new probabilistic model for learning of coupled dynamical systems in latent state spaces. The coupling is achieved by combining predictions from several Gaussian process dynamical models in a product-of-experts fashion. Our approach facilitates modulation of coupling strengths without the need for computationally expensive re-learning of the dynamical models. We demonstrate the effectiveness of the new coupling model on synthetic toy examples and on high-dimensional human walking motion capture data.