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In this paper, we present a new adaptive feature scaling scheme for ultrahigh-dimensional feature selection on Big Data, and then reformulate it as a convex semi-infinite programming (SIP) problem. To address the SIP, we propose an efficient feature generating paradigm. Different from traditional gradient-based approaches that conduct optimization on all input features, the proposed paradigm iteratively activates a group of features, and solves a sequence of multiple kernel learning (MKL) subproblems. To further speed up the training, we propose to solve the MKL subproblems in their primal forms through a modified accelerated proximal gradient approach. Due to such optimization scheme, some efficient cache techniques are also developed. The feature generating paradigm is guaranteed to converge globally under mild conditions, and can achieve lower feature selection bias. Moreover, the proposed method can tackle two challenging tasks in feature selection: 1) group-based feature selection with complex structures, and 2) nonlinear feature selection with explicit feature mappings. Comprehensive experiments on a wide range of synthetic and real-world data sets of tens of million data points with O(10^14) features demonstrate the competitive performance of the proposed method over state-of-the-art feature selection methods in terms of generalization performance and training efficiency.

Motivated by big data applications, first-order methods have been extremely popular in recent years. However, naive gradient methods generally converge slowly. Hence, much efforts have been made to accelerate various first-order methods. This paper proposes two accelerated methods towards solving structured linearly constrained convex programming, for which we assume composite convex objective that is the sum of a differentiable function and a possibly nondifferentiable one. The first method is the accelerated linearized augmented Lagrangian method (LALM). At each update to the primal variable, it allows linearization to the differentiable function and also the augmented term, and thus it enables easy subproblems. Assuming merely convexity, we show that LALM owns O(1/t) convergence if parameters are kept fixed during all the iterations and can be accelerated to O(1/t^2) if the parameters are adapted, where t is the number of total iterations. The second method is the accelerated linearized alternating direction method of multipliers (LADMM). In addition to the composite convexity, it further assumes two-block structure on the objective. Different from classic ADMM, our method allows linearization to the objective and also augmented term to make the update simple. Assuming strong convexity on one block variable, we show that LADMM also enjoys O(1=t2) convergence with adaptive parameters. This result is a significant improvement over that in [Goldstein et. al, SIIMS'14], which requires strong convexity on both block variables and no linearization to the objective or augmented term. Numerical experiments are performed on quadratic programming, image denoising, and support vector machine. The proposed accelerated methods are compared to nonaccelerated ones and also existing accelerated methods. The results demonstrate the validness of acceleration and superior performance of the proposed methods over existing ones.

Inspired by the graph Laplacian and the point integral method, we introduce a novel weighted graph Laplacian method to compute a smooth interpolation function on a point cloud in high dimensional space. The numerical results in semi-supervised learning and image inpainting show that the weighted graph Laplacian is a reliable and efficient interpolation method. In addition, it is easy to implement and faster than graph Laplacian.

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