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We provide an algorithmic framework for structured sparse recovery which unifies combinatorial optimization with the non-smooth convex optimization framework by Nesterov [1, 2]. Our algorithm, dubbed Nesterov iterative hard-thresholding (NIHT), is similar to the algebraic pursuits (ALPS) in [3] in spirit: we use the gradient information in the convex data error objective to navigate over the non convex set of structured sparse signals. While ALPS feature a priori approximation guarantees, we were only able to provide an online approximation guarantee for NIHT (e.g., the guarantees require the algorithm execution). Experiments show however that NIHT can empirically outperform ALPS and other state-of-the-art convex optimization-based algorithms in sparse recovery.