In this paper, we first study $\ell_q$ minimization and its associated iterative reweighted algorithm for recovering sparse vectors. Unlike most existing work, focus on unconstrained minimization, which show a few advantages noisy measurements and/or approximately Inspired by the results in [Daubechies et al., Comm. Pure Appl. Math., 63 (2010), pp. 1--38] constrained start with preliminary yet novel analysis includes convergence, error bound, local convergence behavior. Then, are extended to...

We study an unconstrained version of the $\ell_q$ minimization for sparse solution underdetermined linear systems $0<q\leq1$. Although is nonconvex when $q<1$, we introduce a regularization and develop iterative algorithm. show that algorithm converges solutions converge to under some additional assumptions on systems. Numerical experiments are presented demonstrate effectiveness our approach.

In this paper, we propose an efficient and scalable low rank matrix completion algorithm. The key idea is to extend the orthogonal matching pursuit method from vector case case. We further economic version of our algorithm by introducing a novel weight updating rule reduce time storage complexity. Both versions are computationally inexpensive for each iteration find satisfactory results in few iterations. Another advantage proposed that it has only one tunable parameter, which rank. It easy...

This paper studies the long-existing idea of adding a nice smooth function to "smooth" nondifferentiable objective in context sparse optimization, particular, minimization $\|\mathbf{x}\|_1+\frac{1}{2\alpha}\|\mathbf{x}\|_2^2$, where $\mathbf{x}$ is vector, as well $\|\mathbf{X}\|_*+\frac{1}{2\alpha}\|\mathbf{X}\|_F^2$, $\mathbf{X}$ matrix and $\|\mathbf{X}\|_*$ $\|\mathbf{X}\|_F$ are nuclear Frobenius norms $\mathbf{X}$, respectively. We show that they let vectors low-rank matrices be...

In this paper the asymptotic behavior of penalized spline estimators is studied using bivariate splines over triangulations and an energy functional as penalty. The rate L2 convergence derived, which achieves optimal nonparametric established by Stone (1982). normality established, shown to hold uniformly points where regression function estimated. size conditional variance also evaluated a simple expression for given. Simulation experiments have provided strong evidence that corroborates...

This paper introduces a parallel and distributed extension to the alternating direction method of multipliers (ADMM) for solving convex problem: minimize $\sum_{i=1}^N f_i(x_i)$ subject A_i x_i=c, x_i\in \mathcal{X}_i$. The algorithm decomposes original problem into N smaller subproblems solves them in at each iteration. Jacobian-type is well suited computing particularly attractive certain large-scale problems. few novel results. Firstly, it shows that extending ADMM straightforwardly from...

Surface-based data are prevalent across diverse practical applications in various fields. This paper introduces a novel nonparametric method to discover the underlying signals from distributed on complex surface-based domains. The proposed approach involves penalised spline estimator defined triangulation of surface patches, enabling effective signal extraction and recovery. offers superior handling 'leakage' or 'boundary effects' over domains, enhanced computational efficiency, capabilities...

\textit{Drosophila melanogaster} has been established as a model organism for investigating the fundamental principles of developmental gene interactions. The expression patterns can be documented digital images, which are annotated with anatomical ontology terms to facilitate pattern discovery and comparison. automated annotation images received increasing attention due recent expansion image database. effectiveness relies on quality feature representation. Previous studies have...

It is known that generalized barycentric coordinates can be used to form Bernstein polynomial-like functions over a polygon with any number of sides. We propose use these space continuous polygonal splines (piecewise defined functions) order $d$ partition consisting polygons which able reproduce all polynomials degree $d$. Locally supported basis for the are constructed $d\ge 2$. The construction $d=2$ simpler than "serendipity" quadratic finite elements have appeared in recent literature....

We give many examples of bivariate nonseparable compactly supported orthonormal wavelets whose scaling functions are over [0,3]x[0,3]. The Holder continuity properties these studied.

We consider the functional linear regression model where explanatory variable is a random surface and response real variable, in various situations both noise can be unbounded dependent. Bivariate splines over triangulations represent surfaces. use this representation to construct least squares estimators of function with penalisation term. Under assumptions that regressors sample span large enough space functions, bivariate approximation properties yield consistency estimators. Simulations...

Macro-elements of arbitrary smoothness are constructed on Powell-Sabin triangle splits. These elements useful for solving boundary-value problems and interpolation Hermite data. It is shown that they optimal with respect to spline degree, we believe also the number degrees freedom. The construction provides local bases certain superspline spaces defined over refinements. be stable as a function smallest angle in triangulation, which turn implies associated have order approximation power.

We study minimal energy interpolation and discrete penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous Our show that splines have certain advantages over

Bivariate splines with various degrees are considered in this paper. A matrix form of the extended smoothness conditions for these is presented. Upon form, multivariate spline method numerical solution partial differential equations (PDEs) proposed by Awanou, Lai, and Wenston [The scattered data fitting solutions equations, Wavelets Splines, G. Chen M. J. eds., Nashboro Press, Brentwood, TN, 2006, pp. 24–76] generalized to obtain a new method. It observed that, combined prelocal refinement...