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

Iterative Thresholding for Demixing Structured Superpositions in High Dimensions