Enhanced Random Equivalent Sampling Based on Compressed Sensing

Approximate affine linear relationship between L 1 norm objective functional values and L 2 norm constraint bounds

“…Finally, the third type of experiments is based on signals from experimental measurements. It is found that the relative 2 L norm errors for the first type of experiments, the second type of experiments and the third type of experiments are around 2.3%, 2.6% and 0.2%, respectively. This demonstrates that the relationship between the 1 L norm objective functional values and the 2 L norm specifications of sparse optimization problems is approximately linear.…”

“…Denote ε as the acceptable user defined bound on the 2 L norm of the difference between the original signal and the reconstructed signal. Let * ε x be the optimal solution of the 1 L norm inequality constrained optimization problem.…”

“…[9] are solved instead by the least angle regression (LARS) algorithms or the least absolute shrinkage and selection operator (LASSO) algorithms. However, these optimization problems are different from the original optimization problems (the optimization problems with the 1 L norm objective functions subject to the 2 L norm specifications). Hence, the obtained solutions of these optimization problems are different.…”

“…Although the empirical relationship between the 2 L norm objective functional values and the ∞ L norm specifications has been recently investigated [10], the relationship between the 1 L norm objective functional values and the 2 L norm specifications has not been studied yet. Unlike the optimization problems with the 2 L norm objective functions subject to the 1 L norm specifications where their feasible sets are the convex hulls of a set of vertices and bounded by linear hyperplanes, the feasible sets of the optimization problems with the 1 L norm objective functional values subject to the 2 L norm specifications are not characterized by the convex hulls of a set of vertices and not bounded by linear hyperplanes. Hence, this kind of optimization problems cannot be solved via conventional LARS algorithms or LASSO algorithms.…”

“…The approximate linear relationship between the 1 L norm objective functional values and the 2 L norm specifications is presented in Section 2. A design of the 2 L norm specification is proposed in Section 3. Finally, conclusions are drawn in Section 4.…”