High-Dimensional Data Analysis with Low-Dimensional Models Principles, Computation, and Applications

In the previous chapters, we have studied how either a sparse vector or a low-rank matrix can be recovered from compressive or incomplete measurements. In this chapter, we will show that it is also possible to simultaneously recover a sparse signal and a low-rank signal from their superposition (mixture) or from highly compressive measurements of their superposition (mixture). This combination of rank and sparsity gives rise to a broader class of models that can be used to model richer structures underlying high-dimensional data, as we will see in examples in this chapter and later application chapters. Nevertheless, we are also faced with new technical challenges about whether and how such structures can be recovered correctly and effectively, from few observations.