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We propose an ℓ 1 -penalized estimation procedure for high-dimensional linear mixedeffects models. The models are useful whenever there is a grouping structure among highdimensional observations, that is, for clustered data. We prove a consistency and an oracle optimality result and we develop an algorithm with provable numerical convergence. Furthermore, we demonstrate the performance of the method on simulated and a real high-dimensional data set.

The Ising model is a useful tool for studying complex interactions within a system. The estimation of such a model, however, is rather challenging, especially in the presence of high-dimensional parameters. In this work, we propose efficient procedures for learning a sparse Ising model based on a penalized composite conditional likelihood with nonconcave penalties. Nonconcave penalized likelihood estimation has received a lot of attention in recent years. However, such an approach is computationally prohibitive under high-dimensional Ising models. To overcome such difficulties, we extend the methodology and theory of nonconcave penalized likelihood to penalized composite conditional likelihood estimation. The proposed method can be efficiently implemented by taking advantage of coordinate-ascent and minorization-maximization principles. Asymptotic oracle properties of the proposed method are established with NP-dimensionality. Optimality of the computed local solution is discussed. We demonstrate its finite sample performance via simulation studies and further illustrate our proposal by studying the Human Immunodeficiency Virus type 1 protease structure based on data from the Stanford HIV drug resistance database. Our statistical learning results match the known biological findings very well, although no prior biological information is used in the data analysis procedure.

Uniform Kadec-Klee property, which takes an indispensable part in the researches of some mathematics branches, has attracted increasing extensive exploration and discussion. In this paper, necessary and sufficient conditions for uniform Kadec-Klee property in Orlicz-Lorentz sequence space equipped with Orlicz norm are given.

The necessary and sufficient conditions for both the Kadec–Klee property as well as the Kadec–Klee property with respect to the coordinatewise convergence in Orlicz–Lorentz sequence spaces equipped with the Orlicz norm and generated by arbitrary Orlicz functions as well as any non-increasing weight sequences are given. Moreover, for their subspaces of elements with an order continuous norm the full characterization of the Kadec–Klee property with respect to the coordinatewise convergence is presented. Some tools useful in the proofs of the main results are also provided.

Problems of weight optimization of anisotropic structures with restrictions on strength, stability, etc., are solved by the coordinatewise descent method on a unit interval, combined with effective methods of nonlinear programming (dichotomy and golden section methods).