Convex clustering is a promising new approach to the classical problem of clustering, combining strong performance in empirical studies with rigorous theoretical foundations. Despite these advantages, convex clusterin...
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Convex clustering is a promising new approach to the classical problem of clustering, combining strong performance in empirical studies with rigorous theoretical foundations. Despite these advantages, convex clustering has not been widely adopted, due to its computationally intensive nature and its lack of compelling visualizations. To address these impediments, we introduce algorithmic regularization, an innovative technique for obtaining high-quality estimates of regularization paths using an iterative one-step approximation scheme. We justify our approach with a novel theoretical result, guaranteeing global convergence of the approximate path to the exact solution under easily checked non-data-dependent assumptions. The application of algorithmic regularization to convex clustering yields the Convex Clustering via algorithmic regularization Paths (CARP) algorithm for computing the clustering solution path. On example datasets from genomics and text analysis, CARP delivers over a 100-fold speed-up over existing methods, while attaining a finer approximation grid than standard methods. Furthermore, CARP enables improved visualization of clustering solutions: the fine solution grid returned by CARP can be used to construct a convex clustering-based dendrogram, as well as forming the basis of a dynamic path-wise visualization based on modern web technologies. Our methods are implemented in the open-source R package clustRviz, available at . for this article are available online.
We investigate the statistical behavior of gradient descent iterates with dropout in the linear regression model. In particular, non-asymptotic bounds for the convergence of expectations and covariance matrices of the...
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We investigate the statistical behavior of gradient descent iterates with dropout in the linear regression model. In particular, non-asymptotic bounds for the convergence of expectations and covariance matrices of the iterates are derived. The results shed more light on the widely cited connection between dropout and ℓ2-regularization in the linear model. We indicate a more subtle relationship, owing to interactions between the gradient descent dynamics and the additional randomness induced by dropout. Further, we study a simplified variant of dropout which does not have a regularizing effect and converges to the least squares estimator.
A new algorithm is presented for the numerical integration of second-order ordinary differential equations with perturbations that depend on the first derivative of the dependent variables with respect to the independ...
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A new algorithm is presented for the numerical integration of second-order ordinary differential equations with perturbations that depend on the first derivative of the dependent variables with respect to the independent variable;it is especially designed for few-body problems with velocity-dependent perturbations. The algorithm can be used within extrapolation methods for enhanced accuracy, and it is fully explicit, which makes it a competitive alternative to standard discretization methods.
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