Hierarchical GTM: Constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the parent visualization plot which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 19-dimensional data sets.

Hierarchical GTM: constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping (GTM). bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the ancestor visualization plots which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 18-dimensional data sets.

Constructing localized non-linear projection manifolds in a principled way:hierarchical generative topographic mapping

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis (Bishop98a) in several directions: 1. We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. 2. We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. 3. Using tools from differential geometry we derive expressions for local directionalcurvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the parent visualization plot which are captured by a child model.We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set andapply our system to two more complex 12- and 19-dimensional data sets.

Investigation of neural correlates of faces and emotional expressions using magnetoencephalography

The recognition of faces and of facial expressions in an important evolutionary skill, and an integral part of social communication. It has been argued that the processing of faces is distinct from the processing of non-face stimuli and functional neuroimaging investigations have even found evidence of a distinction between the perception of faces and of emotional expressions. Structural and temporal correlates of face perception and facial affect have only been separately identified. Investigation neural dynamics of face perception per se as well as facial affect would allow the mapping of these in space, time and frequency specific domains. Participants were asked to perform face categorisation and emotional discrimination tasks and Magnetoencephalography (MEG) was used to measure the neurophysiology of face and facial emotion processing. SAM analysis techniques enable the investigation of spectral changes within specific time-windows and frequency bands, thus allowing the identification of stimulus specific regions of cortical power changes. Furthermore, MEG’s excellent temporal resolution allows for the detection of subtle changes associated with the processing of face and non-face stimuli and different emotional expressions. The data presented reveal that face perception is associated with spectral power changes within a distributed cortical network comprising occipito-temporal as well as parietal and frontal areas. For the perception of facial affect, spectral power changes were also observed within frontal and limbic areas including the parahippocampal gyrus and the amygdala. Analyses of temporal correlates also reveal a distinction between the processing of faces and facial affect. Face perception per se occurred at earlier latencies whereas the discrimination of facial expression occurred within a longer time-window. In addition, the processing of faces and facial affect was differentially associated with changes in cortical oscillatory power for alpha, beta and gamma frequencies. The perception of faces and facial affect is associated with distinct changes in cortical oscillatory activity that can be mapped to specific neural structures, specific time-windows and latencies as well as specific frequency bands. Therefore, the work presented in this thesis provides further insight into the sequential processing of faces and facial affect.