Conclusion and Future Work

Abstract

In this thesis, we studied sparsity-constrained optimization problems and proposed a number of greedy algorithms as approximate solvers for these problems. Unlike the existing convex programming methods, the proposed greedy methods do not require the objective to be convex everywhere and produce a solution that is exactly sparse. We showed that if the objective function has well-behaved second order variations, namely if it obeys the SRH or the SRL conditions, then our proposed algorithms provide accurate solutions. Some of these algorithms are also examined through simulations for the 1-bit CS problem and sparse logistic regression. In our work the minimization of functions subject to structured sparsity is also addressed. Assuming the objective function obeys a variant of the SRH condition tailored for model-based sparsity, we showed that a non-convex PGD method can produce an accurate estimate of the underlying parameter.