5 research outputs found

    Non-Convex Methods for Compressed Sensing and Low-Rank Matrix Problems

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    In this thesis we study functionals of the type \mathcal{K}_{f,A,\b}(\x)= \mathcal{Q}(f)(\x) + \|A\x - \b \| ^2 , where AA is a linear map, \b a measurements vector and Q \mathcal{Q} is a functional transform called \emph{quadratic envelope}; this object is a very close relative of the \emph{Lasry-Lions envelope} and its use is meant to regularize the functionals ff. Carlsson and Olsson investigated in earlier works the connections between the functionals \mathcal{K}_{f,A,\b} and their unregularized counterparts f(\x) + \|A\x - \b \| ^2 . For certain choices of ff the penalty Q(f)(β‹…) \mathcal{Q}(f)(\cdot) acts as sparsifying agent and the minimization of \mathcal{K}_{f,A,\b}(\x) delivers sparse solutions to the linear system of equations A\x = \b . We prove existence and uniqueness results of the sparse (or low rank, since the functional ff can have any Hilbert space as domain) global minimizer of \mathcal{K}_{f,A,\b}(\x) for some instances of ff, under Restricted Isometry Property conditions on AA. The theory is complemented with robustness results and a wide range of numerical experiments, both synthetic and from real world problems

    Symmetry and Complexity

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    Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry

    Enhanced Sparsity by Non-Separable Regularization

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