Algorithms for Sparsity-Constrained Optimization by Sohail Bahmani

By Sohail Bahmani

This thesis demonstrates thoughts that supply quicker and extra exact strategies to various difficulties in computer studying and sign processing. the writer proposes a "greedy" set of rules, deriving sparse strategies with promises of optimality. using this set of rules gets rid of some of the inaccuracies that happened with using prior models.

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2011) also proposed a coordinatedescent type algorithm for minimization of a convex and smooth objective over S. 1007/978-3-319-01881-2__3, © Springer International Publishing Switzerland 2014 11 12 3 Sparsity-Constrained Optimization the convex signal/parameter models introduced in Chandrasekaran et al. (2012). This formulation includes the `1 -constrained minimization as a special case, and the algorithm is shown to converge to the minimum in objective value similar to the standard results in convex optimization.

Zhang. Sparse recovery with orthogonal matching pursuit under RIP. IEEE Transactions on Information Theory, 57(9):6215–6221, Sept. 2011. 1 Background Quantization is an indispensable part of digital signal processing and digital communications systems. To incorporate CS methods in these systems, it is thus necessary to analyze and evaluate them considering the effect of measurement quantization. There has been a growing interest in quantized CS in the literature Laska et al. (2009); Dai et al. (2009); Sun and Goyal (2009); Zymnis et al.

1 C à J / mÂmax 24 3 Sparsity-Constrained Optimization Ä k exp ! Q . 7) Note that Assumption (ii) guarantees that ÂQ > 0, and thus the above probability bound will not be vacuous for sufficiently large m. 1C / mÂ Ä Pr > > ; J ÂŒn max jJ jDk ! n Äk exp k o ATJ AJ Q . 1C / m ! Q . 6) that for any x and any k-sparse , ! Q . / mÂh : R Á Q . x/  Ä Á C 4 holds with probability at least 1 ". Thus, the `2 -regularized logistic loss has an SRH constant k Ä 1 C 1C  with probability 1 ". 10. One implication of this result is that for a regime in which k and n grow sufficiently large while nk remains constant one can achieve small failure rates provided that m D ˝ Rk log kn .

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