Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

We first study the error performances of the Vector Weak Rescaled Pure Greedy Algorithm for simultaneous approximation with respect to a dictionary D in a Hilbert space. We show that the convergence rate of the Vector Weak Rescaled Pure Greedy Algorithm on A1(D) and the closure of the convex hull of...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 11; no. 9; p. 2020
Main Authors: Xu, Xu, Guo, Jinyu, Ye, Peixin, Zhang, Wenhui
Format: Journal Article
Language:English
Published: Basel MDPI AG 24.04.2023
Subjects:
Algorithms
Analysis
Approximation
Banach spaces
Chebyshev approximation
Computational geometry
Convergence
Convexity
Dictionaries
Efficiency
Error analysis
Food science
greedy algorithm
Greedy algorithms
Hilbert space
Hilbert spaces
Machine learning
Mathematical analysis
modulus of smoothness
Signal processing
uniformly smooth Banach spaces
vector approximation
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary:We first study the error performances of the Vector Weak Rescaled Pure Greedy Algorithm for simultaneous approximation with respect to a dictionary D in a Hilbert space. We show that the convergence rate of the Vector Weak Rescaled Pure Greedy Algorithm on A1(D) and the closure of the convex hull of the dictionary D is optimal. The Vector Weak Rescaled Pure Greedy Algorithm has some advantages. It has a weaker convergence condition and a better convergence rate than the Vector Weak Pure Greedy Algorithm and is simpler than the Vector Weak Orthogonal Greedy Algorithm. Then, we design a Vector Weak Rescaled Pure Greedy Algorithm in a uniformly smooth Banach space setting. We obtain the convergence properties and error bound of the Vector Weak Rescaled Pure Greedy Algorithm in this case. The results show that the convergence rate of the VWRPGA on A1(D) is sharp. Similarly, the Vector Weak Rescaled Pure Greedy Algorithm is simpler than the Vector Weak Chebyshev Greedy Algorithm and the Vector Weak Relaxed Greedy Algorithm.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math11092020