Dimension reduction is widely used and often necessary to make network analyses and their interpretation tractable by reducing high-dimensional data to a small number of underlying variables. Techniques such as exploratory factor analysis (EFA) are used by neuroscientists to reduce measurements from a large number of brain regions to a tractable number of factors. However, dimension reduction often ignores relevant a priori knowledge about the structure of the data. For example, it is well established that the brain is highly symmetric. In this paper, we (a) show the adverse consequences of ignoring a priori structure in factor analysis, (b) propose a technique to accommodate structure in EFA by using structured residuals (EFAST), and (c) apply this technique to three large and varied brain-imaging network datasets, demonstrating the superior fit and interpretability of our approach. We provide an R software package to enable researchers to apply EFAST to other suitable datasets.