3,993 research outputs found

    Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach

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    We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty lies in finding the projection of a point in the intersection of many sets. Existing approaches yield an infeasible point with an iteration-complexity of O(1/ε2)O(1/\varepsilon^2) for nonsmooth problems with no guarantees on the in-feasibility. By reformulating the problem through exact penalty functions, we derive first-order algorithms which not only guarantees that the distance to the intersection is small but also improve the complexity to O(1/ε)O(1/\varepsilon) and O(1/ε)O(1/\sqrt{\varepsilon}) for smooth functions. For composite and smooth problems, this is achieved through a saddle-point reformulation where the proximal operators required by the primal-dual algorithms can be computed in closed form. We illustrate the benefits of our approach on a graph transduction problem and on graph matching

    Moment-Matching Polynomials

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    We give a new framework for proving the existence of low-degree, polynomial approximators for Boolean functions with respect to broad classes of non-product distributions. Our proofs use techniques related to the classical moment problem and deviate significantly from known Fourier-based methods, which require the underlying distribution to have some product structure. Our main application is the first polynomial-time algorithm for agnostically learning any function of a constant number of halfspaces with respect to any log-concave distribution (for any constant accuracy parameter). This result was not known even for the case of learning the intersection of two halfspaces without noise. Additionally, we show that in the "smoothed-analysis" setting, the above results hold with respect to distributions that have sub-exponential tails, a property satisfied by many natural and well-studied distributions in machine learning. Given that our algorithms can be implemented using Support Vector Machines (SVMs) with a polynomial kernel, these results give a rigorous theoretical explanation as to why many kernel methods work so well in practice

    Existence, regularity and structure of confined elasticae

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    We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set Ω\Omega. We prove existence, regularity and some structural properties of minimizers. In particular, when Ω\Omega is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a minimizer with self-intersections

    Stability and Error Analysis for Optimization and Generalized Equations

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    Stability and error analysis remain challenging for problems that lack regularity properties near solutions, are subject to large perturbations, and might be infinite dimensional. We consider nonconvex optimization and generalized equations defined on metric spaces and develop bounds on solution errors using the truncated Hausdorff distance applied to graphs and epigraphs of the underlying set-valued mappings and functions. In the process, we extend the calculus of such distances to cover compositions and other constructions that arise in nonconvex problems. The results are applied to constrained problems with feasible sets that might have empty interiors, solution of KKT systems, and optimality conditions for difference-of-convex functions and composite functions

    Scalable Algorithms for Tractable Schatten Quasi-Norm Minimization

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    The Schatten-p quasi-norm (0<p<1)(0<p<1) is usually used to replace the standard nuclear norm in order to approximate the rank function more accurately. However, existing Schatten-p quasi-norm minimization algorithms involve singular value decomposition (SVD) or eigenvalue decomposition (EVD) in each iteration, and thus may become very slow and impractical for large-scale problems. In this paper, we first define two tractable Schatten quasi-norms, i.e., the Frobenius/nuclear hybrid and bi-nuclear quasi-norms, and then prove that they are in essence the Schatten-2/3 and 1/2 quasi-norms, respectively, which lead to the design of very efficient algorithms that only need to update two much smaller factor matrices. We also design two efficient proximal alternating linearized minimization algorithms for solving representative matrix completion problems. Finally, we provide the global convergence and performance guarantees for our algorithms, which have better convergence properties than existing algorithms. Experimental results on synthetic and real-world data show that our algorithms are more accurate than the state-of-the-art methods, and are orders of magnitude faster.Comment: 16 pages, 5 figures, Appears in Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI), Phoenix, Arizona, USA, pp. 2016--2022, 201

    Partial separability revisited: Necessary and sufficient criteria

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    We extend the classification of mixed states of quantum systems composed of arbitrary number of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives 1+18+1 partial separability classes in the tripartite case contrary to a former 1+8+1. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.Comment: v3: 22 pages, 5 tables, 1 figure, minor corrections (typos), clarification in the Introduction. Accepted in Phys. Rev. A. Comments are welcom
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