  Brain decoding involves the determination of a subject's cognitive state or
an associated stimulus from functional neuroimaging data measuring brain
activity. In this setting the cognitive state is typically characterized by an
element of a finite set, and the neuroimaging data comprise voluminous amounts
of spatiotemporal data measuring some aspect of the neural signal. The
associated statistical problem is one of classification from high-dimensional
data. We explore the use of functional principal component analysis, mutual
information networks, and persistent homology for examining the data through
exploratory analysis and for constructing features characterizing the neural
signal for brain decoding. We review each approach from this perspective, and
we incorporate the features into a classifier based on symmetric multinomial
logistic regression with elastic net regularization. The approaches are
illustrated in an application where the task is to infer, from brain activity
measured with magnetoencephalography (MEG), the type of video stimulus shown to
a subject.
