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  1. null (Ed.)
    As a popular tool for producing meaningful and interpretable models, large-scale sparse learning works efficiently in many optimization applications when the underlying structures are indeed or close to sparse. However, naively applying the existing regularization methods can result in misleading outcomes because of model misspecification. In this paper, we consider nonsparse learning under the factors plus sparsity structure, which yields a joint modeling of sparse individual effects and common latent factors. A new methodology of nonsparse learning with latent variables (NSL) is proposed for joint estimation of the effects of two groups of features, one for individual effects and the other associated with the latent substructures, when the nonsparse effects are captured by the leading population principal component score vectors. We derive the convergence rates of both sample principal components and their score vectors that hold for a wide class of distributions. With the properly estimated latent variables, properties including model selection consistency and oracle inequalities under various prediction and estimation losses are established. Our new methodology and results are evidenced by simulation and real-data examples. 
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  2. Summary

    High dimensional sparse modelling via regularization provides a powerful tool for analysing large-scale data sets and obtaining meaningful interpretable models. The use of non-convex penalty functions shows advantage in selecting important features in high dimensions, but the global optimality of such methods still demands more understanding. We consider sparse regression with a hard thresholding penalty, which we show to give rise to thresholded regression. This approach is motivated by its close connection with L0-regularization, which can be unrealistic to implement in practice but of appealing sampling properties, and its computational advantage. Under some mild regularity conditions allowing possibly exponentially growing dimensionality, we establish the oracle inequalities of the resulting regularized estimator, as the global minimizer, under various prediction and variable selection losses, as well as the oracle risk inequalities of the hard thresholded estimator followed by further L2-regularization. The risk properties exhibit interesting shrinkage effects under both estimation and prediction losses. We identify the optimal choice of the ridge parameter, which is shown to have simultaneous advantages to both the L2-loss and the prediction loss. These new results and phenomena are evidenced by simulation and real data examples.

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