  Principal component analysis (PCA) is an exploratory tool widely used in data
analysis to uncover dominant patterns of variability within a population.
Despite its ability to represent a data set in a low-dimensional space, the
interpretability of PCA remains limited. However, in neuroimaging, it is
essential to uncover clinically interpretable phenotypic markers that would
account for the main variability in the brain images of a population. Recently,
some alternatives to the standard PCA approach, such as Sparse PCA, have been
proposed, their aim being to limit the density of the components. Nonetheless,
sparsity alone does not entirely solve the interpretability problem, since it
may yield scattered and unstable components. We hypothesized that the
incorporation of prior information regarding the structure of the data may lead
to improved relevance and interpretability of brain patterns. We therefore
present a simple extension of the popular PCA framework that adds structured
sparsity penalties on the loading vectors in order to identify the few stable
regions in the brain images accounting for most of the variability. Such
structured sparsity can be obtained by combining l1 and total variation (TV)
penalties, where the TV regularization encodes higher order information about
the structure of the data. This paper presents the structured sparse PCA
(denoted SPCA-TV) optimization framework and its resolution. We demonstrate the
efficiency and versatility of SPCA-TV on three different data sets. The gains
of SPCA-TV over unstructured approaches are significant,since SPCA-TV reveals
the variability within a data set in the form of intelligible brain patterns
that are easy to interpret, and are more stable across different samples.
