Almost Optimality of the Orthogonal Super Greedy Algorithm for μ-Coherent Dictionaries

Shao, Cheng-Gang; Chang, Jincai; Ye, Peixin; Zhang, Wenhui; Xing, Shuo

doi:10.3390/axioms11050186

Axioms

2022

DOI: 10.3390/axioms11050186

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Almost Optimality of the Orthogonal Super Greedy Algorithm for μ-Coherent Dictionaries

Cheng-Gang Shao

Jincai Chang

Peixin Ye

et al.

Abstract: We study the approximation capability of the orthogonal super greedy algorithm (OSGA) with respect to μ-coherent dictionaries in Hilbert spaces. We establish the Lebesgue-type inequalities for OSGA, which show that the OSGA provides an almost optimal approximation on the first [1/(18μs)] steps. Moreover, we improve the asymptotic constant in the Lebesgue-type inequality of OGA obtained by Livshitz E D.

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2024

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Cited by 3 publications

References 18 publications

(20 reference statements)

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Optimality of the rescaled pure greedy learning algorithms

Zhang

Xing

2022

Int. J. Wavelets Multiresolut Inf. Process.

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We propose the Rescaled Pure Greedy Learning Algorithm (RPGLA) for solving the kernel-based regression problem. The computational complexity of the RPGLA is less than the Orthogonal Greedy Learning Algorithm (OGLA) and Relaxed Greedy Learning Algorithm (RGLA). We obtain the convergence rates of the RPGLA for continuous kernels. When the kernel is infinitely smooth, we derive a convergence rate that can be arbitrarily close to the best rate [Formula: see text] under a mild assumption of the regression function.

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Optimality of the rescaled pure greedy learning algorithms

Zhang

Xing

2022

Int. J. Wavelets Multiresolut Inf. Process.

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Optimality of the Approximation and Learning by the Rescaled Pure Super Greedy Algorithms

Zhang

Xing

et al. 2022

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We propose the Weak Rescaled Pure Super Greedy Algorithm (WRPSGA) for approximation with respect to a dictionary 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 μ-coherent dictionary 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(m−1) under some mild assumptions.

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Sparse Signal Recovery via Rescaled Matching Pursuit

Li,

2024

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We propose the Rescaled Matching Pursuit (RMP) algorithm to recover sparse signals in high-dimensional Euclidean spaces. The RMP algorithm has less computational complexity than other greedy-type algorithms, such as Orthogonal Matching Pursuit (OMP). We show that if the restricted isometry property is satisfied, then the upper bound of the error between the original signal and its approximation can be derived. Furthermore, we prove that the RMP algorithm can find the correct support of sparse signals from random measurements with a high probability. Our numerical experiments also verify this conclusion and show that RMP is stable with the noise. So, the RMP algorithm is a suitable method for recovering sparse signals.

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Almost Optimality of the Orthogonal Super Greedy Algorithm for μ-Coherent Dictionaries

Cited by 3 publications

References 18 publications

Optimality of the rescaled pure greedy learning algorithms

Optimality of the rescaled pure greedy learning algorithms

Optimality of the Approximation and Learning by the Rescaled Pure Super Greedy Algorithms

Sparse Signal Recovery via Rescaled Matching Pursuit

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