Analysis of electro-encephalographic (EEG) signals has been proven an extremely useful research tool for studying the neural correlates of behavior. Single-trial analysis of these signals is essential for the development of non-invasive Brain Computer Interfaces. Analysis of these signals is often expressed as a single-trial classification problem. The goal is to infer the underling cognitive state of an individual using purely EEG signals. The high dimensional space of EEG observations and the low signal-to-noise ratio (SNR) - often -20db or less - as well as the inter-subject variability and limited observations available for training, make the single-trial classification of EEG an extremely challenging computational task. To address these challenges we introduce concepts from Multi-linear Algebra and incorporate EEG domain knowledge. More precisely, we formulate the problem in a matrix space and introduce a bilinear combination of a matrix to reduce the space dimensions. Thus the title of this dissertation: "The Bilinear Brain". We also address the issue of inter-subject variability by defining a model that is partially subject-invariant. We develop two classification algorithms based on the Bilinear model. We term the first algorithm Second Order Bilinear Discriminant Analysis (SOBDA). It combines first order and second order statistics of the observation space. The second algorithm we term Bilinear Feature Based Discriminant (BFBD) and addresses the issue of inter-subject variability. We evaluate our methods on both simulated and real human EEG data-sets and show that our method outperforms state-of-the-art methods on different experimental paradigms.
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