| The behavior of the human brain has long puzzled researchers in diverse fields including biology, psychology, and engineering. The emerging fields of compressed sensing (CS) and sparse approximation (SA) provide new tools for analyzing large data sets generated by the brain. A number of techniques which seek sparse solutions to data fitting problems, including the well known LASSO, have been developed in CS and SA. The Group LASSO (GLASSO) extends the LASSO to problems in which sparse groups of coefficients are used to represent the data. The GLASSO is often solved using second order cone programming (SOCP).;In this thesis, an expectation-maximization (EM) algorithm approach to group sparsity-constrained optimization problems is derived. The EM formulation allows efficient calculation of minimizers for a range of regularization parameters and with a variety of group penalties, including the ℓ2 norm of each group, i.e. the GLASSO. The computational complexity of the EM approach scales linearly with problem size, vs. quadratic scaling for SOCP. This EM algorithm approach to the GLASSO is then applied to two main challenges in brain research.;First, the GLASSO is used in the MEG/EEG inverse problem to identify a small number of space-time events which fit the measured data using a novel spatio-temporal decomposition of cortical signals. The approach is demonstrated with spatial basis derived from overlapping cortical patches, though other spatial bases are easily incorporated.;Second, a modified GLASSO is applied to multivariate autoregressive (MAR) network estimation. This approach seeks neurological networks which have a small number of connections, but no restrictions on the coefficients describing each connection. Conditions which dictate asymptotically consistent estimation of sparse networks are derived and their consequences examined on example networks. In the modified GLASSO approach, self-connections are not penalized. This change improves the ability to recover many networks, and often reduces the tendency to reverse the direction of causality. Exploiting sparse connectivity allows fewer samples to be used in MAR network estimation.;Both the ESP approach to the MEG/EEG inverse problem and the modified GLASSO for MAR network estimation are demonstrated on simulated and real data sets. These approaches provide new tools for understanding the complex function of the human brain. |
