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While matrix-covariate regression models have been studied in many existing works, classical statistical and computational methods for the analysis of the regression coefficient estimation are highly affected by high dimensional matrix-valued covariates. To address these issues, this paper proposes a framework of matrix-covariate regression models based on a low-rank constraint and an additional regularization term for structured signals, with considerations of models of both continuous and binary responses. We propose an efficient Riemannian-steepest-descent algorithm for regression coefficient estimation. We prove that the consistency of the proposed estimator is in the order of O(sqrt{r(q+m)+p}/sqrt{n}), where r is the rank, p x m is the dimension of the coefficient matrix and p is the dimension of the coefficient vector. When the rank r is small, this rate improves over O(sqrt{qm+p}/sqrt{n}), the consistency of the existing work (Li et al. in Electron J Stat 15:1909-1950, 2021) that does not apply a rank constraint. In addition, we prove that all accumulation points of the iterates have similar estimation errors asymptotically and substantially attaining the minimax rate. We validate the proposed method through a simulated dataset on two-dimensional shape images and two real datasets of brain signals and microscopic leucorrhea images. 

Surface wrinkles have emerged as a promising avenue for the development of smart adhesives with dynamically tunable adhesion, finding applications in diverse fields, such as soft robots and medical devices. Despite intensive studies and great achievements, it is still challenging to model and simulate the tunable adhesion with surface wrinkles due to roughened surface topologies and pre-stress inside the materials. The lack of a mechanistic understanding hinders the rational design of these smart adhesives. Here, we integrate a lattice model for nonlinear deformations of solids and nonlocal interaction potentials for adhesion in the framework of molecular dynamics to explore the roles of surface wrinkles on adhesion behaviors. We validate the proposed model by comparing wrinkles in a neo-Hookean bilayer with benchmarked results and reproducing the analytical solution for cylindrical adhesion. We then systematically study the pull-off force of the wrinkled surface with varied compressive strains and adhesion energies. Our results reveal the competing effect between the adhesion-induced contact and the roughness due to wrinkles on enhancing or weakening the adhesion. Such understanding provides guidance for tailoring material and geometry as well as loading wrinkled surfaces for different applications.