The Discriminative Discrete Basis Problem: Definitions, Algorithms, Benchmarking, and Application to Brain's Functional Dynamics

In this paper, considering two Boolean matrices, we tackle the problem of finding a set of Boolean basis vectors that capture the latent components most discriminative of one matrix from another. We refer to this problem as the discriminative discrete basis problem (DDBP). Depending on the symmetry/asymmetry of the problem about the two input matrices, different classes of the DDBP are defined. We present an ensemble of new algorithms, called the discriminative-associative (DASSO) algorithms, to find solutions for these classes of the DDBP. Furthermore, we design a comprehensive, simulation-based benchmarking framework for evaluating the performance of such algorithmic solutions to the DDBP over a wide range of variables, including the effects of disturbances, noise, the geometric features of the latent components within each matrix, and the relationships among these components across the contrasted matrices. As an example application, we demonstrate the utility of the DASSO algorithms in studying the dynamics of the brain function by identifying the cortical activities that discriminate motor from non-motor tasks using recordings obtained via electroencephalography (EEG). The results highlight the location of such discriminating activities in the cortex and the time of their occurrence, which can lead to the design of efficient and accurate brain computer interfaces (BCIs). To the best of our knowledge, this is the first time the DDBP is recognized, defined, and addressed through the development of algorithmic solutions. The DASSO algorithms can be utilized in a variety of data mining and feature selection applications, in which finding discriminating structures across two datasets is of interest.