We propose the Weak Rescaled Pure Super Greedy Algorithm (WRPSGA) for approximation with respect to a dictionary $\mathcal{D}$ in Hilbert space. The WRPSGA is simpler than some popular greedy algorithms. We show that the convergence rate of the RPSGA on the closure of the convex hull of the $\mu$-coherent dictionary $\mathcal{D}$ is optimal. Then, we design the Rescaled Pure Super Greedy Learning Algorithm (RPSGLA) for kernel-based supervised learning. We prove that the convergence rate of the RPSGLA can be arbitrarily close to the best rate $O\left(m^{-1}\right)$ under some mild assumptions.

We construct the blending-type modified Bernstein–Durrmeyer operators and investigate their approximation properties. First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. Finally, we derive the rate of convergence for functions with a derivative of bounded variation. The results show that the new operators have good approximation properties.

We derive the convergence rate of the moving least-squares learning algorithm for regression under the assumption that the samples are drawn from a non-identical sequence of probability measures. The error analysis is carried out by analysing the drift error and using the probability inequalities for the non-identical sampling. When the sequence of marginal distributions converges exponentially to marginal distribution in the dual of a Hölder space, we obtain the satisfactory capacity dependent error bounds of the algorithm that can be arbitrarily close to the rate $O\left(m^{-1}\right)$.

We propose the Group Orthogonal Matching Pursuit (GOMP) algorithm to recover group sparse signals from noisy measurements. Under the group restricted isometry property (GRIP), we prove the instance optimality of the GOMP algorithm for any decomposable approximation norm. Meanwhile, we show the robustness of the GOMP under the measurement error. Compared with the P-norm minimization approach, the GOMP is easier to implement, and the assumption of $\gamma$-decomposability is not required. The simulation results show that the GOMP is very efficient for group sparse signal recovery and significantly outperforms Basis Pursuit in both scalability and solution quality.