Iterative Thresholding for Demixing Structured Superpositions in High Dimensions

We consider the demixing problem of two (or more) high-dimensional vectors from nonlinear observations when the number of such observations is far less than the ambient dimension of the underlying vectors. Specifically, we demonstrate an algorithm that stably estimate the underlying components under general \emph{structured sparsity} assumptions on these components. Specifically, we show that for certain types of structured superposition models, our method provably recovers the components given merely $n = \mathcal{O}(s)$ samples where $s$ denotes the number of nonzero entries in the underlying components. Moreover, our method achieves a fast (linear) convergence rate, and also exhibits fast (near-linear) per-iteration complexity for certain types of structured models. We also provide a range of simulations to illustrate the performance of the proposed algorithm.