Hilbert Space Embedding of Distributions
What is interesting about a distribution is often its behavior when
taking expectations. In this work, distributions are embedded into Hilbert spaces via the averages or mean maps, and then all subsequent operations on distributions are carried out in the Hilbert space.
This allows us to compute distances between distributions in
terms of distances between averages. We have developed a framework of learning based on this which includes
density estimation, clustering, feature selection, two sample tests,
independence tests, nonparametric sorting, and dimensional reduction. A large number of existing methods appear in this framework as special cases.
Furthermore, this often leads to algorithms which are simpler and more effective than information theoretic methods in a broad range of applications.
Modeling, Analyzing and Visualizing Networks
Much of the world's information has a relational structure and can be modelled mathematically as networks and graphs. Examples include biological networks, webgraphs and social networks. Many of these large and complex networks exhibit rich spatial and temporal phenomena. Traditional graph modeling, analysis and visualization algorithms are not able to capture this complex spatial and temporal behavior. We designed new modeling, analyzing and visualizing tools to better understand complex networks.
Biomedical Signal Processing
Brain-computer interface (BCI) is a communication system that relies on the brain rather than the body for control and feedback. My research employs a novel type of features based explicitly on the neurophysiology of EEG signals for classification. Basically, EEG signals are considered as the outputs of a networked dynamical system. The nodes of this system consist of cortical patches, while the links correspond to neural fibers. A large and complex system like this often generates interesting collective dynamics, such as synchronization in the activities of the nodes, and they result in the change of EEG patterns measured on the scalp. These features from the collective dynamics of the system are employed for classification.