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The regression problem of modeling several response variables using the same set of input variables is considered. The model is linearly parameterized and the parameters are estimated by minimizing the error sum of squares subject to a sparsity constraint. The constraint has the effect of eliminating useless inputs and constraining the parameters of the remaining inputs in the model. Two algorithms for solving the resulting convex cone programming problem are proposed. The first algorithm gives a pointwise solution, while the second one computes the entire path of solutions as a function of the constraint parameter. Based on experiments with real data sets, the proposed method has a similar performance to existing methods. In simulation experiments, the proposed method is competitive both in terms of prediction accuracy and correctness of input selection. The advantages become more apparent when many correlated inputs are available for model construction.

Background: DNA amplifications alter gene dosage in cancer genomes by multiplying the gene copy number. Amplifications are quintessential in a considerable number of advanced cancers of various anatomical locations. The aims of this study were to classify human cancers based on their amplification patterns, explore the biological and clinical fundamentals behind their amplificationpattern based classification, and understand the characteristics in human genomic architecture that associate with amplification mechanisms.

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Input selection is advantageous in regression problems. For example, it might decrease the training time of models, reduce measurement costs, and circumvent problems of high dimensionality. Inclusion of useless inputs into the model increases also the likelihood of overfitting. Neural networks provide good generalization in many cases, but their interpretability is usually limited. However, selecting a subset of variables and estimating their relative importances would be valuable in many real world applications. In the present work, a simultaneous input and basis function selection method for the radial basis function (RBF) network is proposed. The selection is performed by minimizing a constrained cost function, in which sparsity of the network is controlled by two continuous valued shrinkage parameters. Each input dimension is weighted and the constraints are imposed on these weights and the output layer coefficients. Direct and alternating optimization procedures are presented to solve the problem. The proposed method is applied to simulated and benchmark data. In the comparison with existing methods, the resulting RBF networks have similar prediction accuracies with the smaller numbers of inputs and basis functions.