750 research outputs found

    Does generalization performance of lql^q regularization learning depend on qq? A negative example

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    lql^q-regularization has been demonstrated to be an attractive technique in machine learning and statistical modeling. It attempts to improve the generalization (prediction) capability of a machine (model) through appropriately shrinking its coefficients. The shape of a lql^q estimator differs in varying choices of the regularization order qq. In particular, l1l^1 leads to the LASSO estimate, while l2l^{2} corresponds to the smooth ridge regression. This makes the order qq a potential tuning parameter in applications. To facilitate the use of lql^{q}-regularization, we intend to seek for a modeling strategy where an elaborative selection on qq is avoidable. In this spirit, we place our investigation within a general framework of lql^{q}-regularized kernel learning under a sample dependent hypothesis space (SDHS). For a designated class of kernel functions, we show that all lql^{q} estimators for 0<q<∞0< q < \infty attain similar generalization error bounds. These estimated bounds are almost optimal in the sense that up to a logarithmic factor, the upper and lower bounds are asymptotically identical. This finding tentatively reveals that, in some modeling contexts, the choice of qq might not have a strong impact in terms of the generalization capability. From this perspective, qq can be arbitrarily specified, or specified merely by other no generalization criteria like smoothness, computational complexity, sparsity, etc..Comment: 35 pages, 3 figure

    Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations

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    In this work, we apply stochastic collocation methods with radial kernel basis functions for an uncertainty quantification of the random incompressible two-phase Navier-Stokes equations. Our approach is non-intrusive and we use the existing fluid dynamics solver NaSt3DGPF to solve the incompressible two-phase Navier-Stokes equation for each given realization. We are able to empirically show that the resulting kernel-based stochastic collocation is highly competitive in this setting and even outperforms some other standard methods
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