Classical unsupervised algorithms, such as k-means and PCA, utilize a simple generativemodel where the sampling distribution is determined by a collection of unobserved, latent
features. While this paradigm is powerful, it has the following consequence for applied
settings: any structured trend in the data must be explained by the latent features and the
assumptions therein. This requirement complicates the analysis of structured data sources,
such as images, videos, and networks, especially when the latent features of interest do not
govern every structured aspect of the data.

In this work, we consider scenarios where the latent features may be partially decoupled from the structure of the data. Under this new setting we develop new algorithmicimprovements and insights for the following problems:
• Tissue intensity recovery for contaminated MRIs, where each pixel intensity is determined by an underlying tissue type and a spatially varying gain field.
• Semi-supervised node classification with graph aggregated features, where nodes are
assumed to follow a community-based structure