RSP-Based Analysis for Sparsest and Least ℓ1\ell_1-Norm Solutions to Underdetermined Linear Systems

Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does $\ell_1$-minimization find the sparsest solution to an underdetermined linear system? In this paper, a deterministic analysis, rooted in the classic linear programming theory, is carried out to further address this question. We first identify a necessary and sufficient condition for the uniqueness of least $\ell_1$-norm solutions to linear systems. From this condition, we deduce that a sparsest solution coincides with the unique least $\ell_1$-norm solution to a linear system if and only if the so-called \emph{range space property} (RSP) holds at this solution. This yields a broad understanding of the relationship between $\ell_0$- and $\ell_1$-minimization problems. Our analysis indicates that the RSP truly lies at the heart of the relationship between these two problems. Through RSP-based analysis, several important questions in this field can be largely addressed. For instance, how to efficiently interpret the gap between the current theory and the actual numerical performance of $\ell_1$-minimization by a deterministic analysis, and if a linear system has multiple sparsest solutions, when does $\ell_1$-minimization guarantee to find one of them? Moreover, new matrix properties (such as the \emph{RSP of order $K$} and the \emph{Weak-RSP of order $K$}) are introduced in this paper, and a new theory for sparse signal recovery based on the RSP of order $K$ is established

Existence of A Solution to Nonlinear Variational Inequality under Generalized Positive Homogeneity

We establish several new existence theorems for nonlinear variational inequalities with generalized positively homogeneous functions. The results presented here are general enough to include two Moré existence theorems of complementarity problems as the special cases. We also establish an existence result for the nonlinear complementarity problem with an exceptional regularity map. The concept of exceptional family for variational inequality plays a key role in our analysis

Alternative Theorems for Nonlinear Projection Equations and Applications to Generalized Complementarity Problems

. This paper introduces two concepts of exceptional families for a class of nonlinear projection equations which provide a unified formulation of several special cases such as finite-dimensional variational inequalities and complementarity problems. Based on these concepts, two alternative theorems, and then two sufficient existence conditions, are established for this class of nonlinear projection equations. We use one of the alternative theorems to establish several new existence results for generalized complementarity problems with new classes of functions such as quasi-P , and P (ø; ff; fi)-maps. The class of quasi-P -maps is broad enough to encompass the union of quasi-monotone maps and nonlinear P -maps. The existence results established here significantly relax several previous solution conditions in the literature. Key words. Nonlinear projection equations, variational inequalities, generalized complementarity problems, exceptional families. 1 The research of this author i..

root length data

root length dat

Marine Mixed Layer Height Detection Using Ship-Borne Coherent Doppler Wind Lidar Based on Constant Turbulence Threshold

Marine mixed layer height (MLH) detection using a ship-borne coherent Doppler wind lidar (CDWL) based on a constant turbulent kinetic energy dissipation rate (TKEDR) threshold is realized and experimentally demonstrated. The MLH can be retrieved from the TKEDR estimated by the CDWL via setting an appropriate threshold. Here, the value of threshold is determined by a reference MLH retrieved from aerosol backscattered signal. The threshold of 10−4 m2 s−3 is found to be applicable in retrieving both inland and marine MLHs. In the experiments, to validate the reliability of the constant threshold, the MLH diurnal cycles at inland and marine sites are retrieved by using a ground-based CDWL. The MLH retrieval result at the marine site shows good agreement with radiosonde-derived MLH. After that, by using a ship-borne CDWL, the marine MLH along the ship’s route in South China Sea is successfully detected in real time

biomass data

biomass dat