Learning with sparsity and uncertainty by Difference of Convex functions optimization

Abstract : In this thesis, we focus on developing optimization approaches for solving some classes of optimization problems in sparsity and robust optimization for data uncertainty. Our methods are based on DC (Difference of Convex functions) programming and DCA (DC Algorithms) which are well-known as powerful tools in optimization. This thesis is composed of two parts: the first part concerns with sparsity while the second part deals with uncertainty. In the first part, a unified DC approximation approach to optimization problem involving the zero-norm in objective is thoroughly studied on both theoretical and computational aspects. We consider a common DC approximation of zero-norm that includes all standard sparse inducing penalty functions, and develop general DCA schemes that cover all standard algorithms in the field. Next, the thesis turns to the nonnegative matrix factorization (NMF) problem. We investigate the structure of the considered problem and provide appropriate DCA based algorithms. To enhance the performance of NMF, the sparse NMF formulations are proposed. Continuing this topic, we study the dictionary learning problem where sparse representation plays a crucial role. In the second part, we exploit robust optimization technique to deal with data uncertainty for two important problems in machine learning: feature selection in linear Support Vector Machines and clustering. In this context, individual data point is uncertain but varies in a bounded uncertainty set. Different models (box/spherical/ellipsoidal) related to uncertain data are studied. DCA based algorithms are developed to solve the robust problems
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Xuan Thanh Vo. Learning with sparsity and uncertainty by Difference of Convex functions optimization. Other [cs.OH]. Université de Lorraine, 2015. English. ⟨NNT : 2015LORR0193⟩. ⟨tel-01754450⟩

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